Next: Higher moments (skewness and kurtosis), Previous: Mean and standard deviation and variance, Up: Statistics [Index]
This function computes the absolute deviation from the mean of data, a dataset of length n with stride stride. The absolute deviation from the mean is defined as,
absdev = (1/N) \sum |x_i - \Hat\mu|
where x_i are the elements of the dataset data. The
absolute deviation from the mean provides a more robust measure of the
width of a distribution than the variance. This function computes the
mean of data via a call to gsl_stats_mean.
This function computes the absolute deviation of the dataset data relative to the given value of mean,
absdev = (1/N) \sum |x_i - mean|
This function is useful if you have already computed the mean of data (and want to avoid recomputing it), or wish to calculate the absolute deviation relative to another value (such as zero, or the median).
Next: Sorting Eigenvalues and Eigenvectors, Previous: Complex Generalized Hermitian-Definite Eigensystems, Up: Eigensystems [Index]
Given two square matrices (A, B), the generalized nonsymmetric eigenvalue problem is to find eigenvalues \lambda and eigenvectors x such that
A x = \lambda B x
We may also define the problem as finding eigenvalues \mu and eigenvectors y such that
\mu A y = B y
Note that these two problems are equivalent (with \lambda = 1/\mu) if neither \lambda nor \mu is zero. If say, \lambda is zero, then it is still a well defined eigenproblem, but its alternate problem involving \mu is not. Therefore, to allow for zero (and infinite) eigenvalues, the problem which is actually solved is
\beta A x = \alpha B x
The eigensolver routines below will return two values \alpha and \beta and leave it to the user to perform the divisions \lambda = \alpha / \beta and \mu = \beta / \alpha.
If the determinant of the matrix pencil A - \lambda B is zero for all \lambda, the problem is said to be singular; otherwise it is called regular. Singularity normally leads to some \alpha = \beta = 0 which means the eigenproblem is ill-conditioned and generally does not have well defined eigenvalue solutions. The routines below are intended for regular matrix pencils and could yield unpredictable results when applied to singular pencils.
The solution of the real generalized nonsymmetric eigensystem problem for a matrix pair (A, B) involves computing the generalized Schur decomposition
A = Q S Z^T B = Q T Z^T
where Q and Z are orthogonal matrices of left and right Schur vectors respectively, and (S, T) is the generalized Schur form whose diagonal elements give the \alpha and \beta values. The algorithm used is the QZ method due to Moler and Stewart (see references).
This function allocates a workspace for computing eigenvalues of n-by-n real generalized nonsymmetric eigensystems. The size of the workspace is O(n).
This function frees the memory associated with the workspace w.
This function sets some parameters which determine how the eigenvalue
problem is solved in subsequent calls to gsl_eigen_gen.
If compute_s is set to 1, the full Schur form S will be
computed by gsl_eigen_gen. If it is set to 0,
S will not be computed (this is the default setting). S
is a quasi upper triangular matrix with 1-by-1 and 2-by-2 blocks
on its diagonal. 1-by-1 blocks correspond to real eigenvalues, and
2-by-2 blocks correspond to complex eigenvalues.
If compute_t is set to 1, the full Schur form T will be
computed by gsl_eigen_gen. If it is set to 0,
T will not be computed (this is the default setting). T
is an upper triangular matrix with non-negative elements on its diagonal.
Any 2-by-2 blocks in S will correspond to a 2-by-2 diagonal
block in T.
The balance parameter is currently ignored, since generalized balancing is not yet implemented.
This function computes the eigenvalues of the real generalized nonsymmetric matrix pair (A, B), and stores them as pairs in (alpha, beta), where alpha is complex and beta is real. If \beta_i is non-zero, then \lambda = \alpha_i / \beta_i is an eigenvalue. Likewise, if \alpha_i is non-zero, then \mu = \beta_i / \alpha_i is an eigenvalue of the alternate problem \mu A y = B y. The elements of beta are normalized to be non-negative.
If S is desired, it is stored in A on output. If T is desired, it is stored in B on output. The ordering of eigenvalues in (alpha, beta) follows the ordering of the diagonal blocks in the Schur forms S and T. In rare cases, this function may fail to find all eigenvalues. If this occurs, an error code is returned.
This function is identical to gsl_eigen_gen except that it also
computes the left and right Schur vectors and stores them into Q
and Z respectively.
This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real generalized nonsymmetric eigensystems. The size of the workspace is O(7n).
This function frees the memory associated with the workspace w.
This function computes eigenvalues and right eigenvectors of the
n-by-n real generalized nonsymmetric matrix pair
(A, B). The eigenvalues are stored in (alpha, beta)
and the eigenvectors are stored in evec. It first calls
gsl_eigen_gen to compute the eigenvalues, Schur forms, and
Schur vectors. Then it finds eigenvectors of the Schur forms and
backtransforms them using the Schur vectors. The Schur vectors are
destroyed in the process, but can be saved by using
gsl_eigen_genv_QZ. The computed eigenvectors are normalized
to have unit magnitude. On output, (A, B) contains
the generalized Schur form (S, T). If gsl_eigen_gen
fails, no eigenvectors are computed, and an error code is returned.
This function is identical to gsl_eigen_genv except that it also
computes the left and right Schur vectors and stores them into Q
and Z respectively.
Next: Sorting Eigenvalues and Eigenvectors, Previous: Complex Generalized Hermitian-Definite Eigensystems, Up: Eigensystems [Index]
Next: Matrices, Previous: Blocks, Up: Vectors and Matrices [Index]
Vectors are defined by a gsl_vector structure which describes a
slice of a block. Different vectors can be created which point to the
same block. A vector slice is a set of equally-spaced elements of an
area of memory.
The gsl_vector structure contains five components, the
size, the stride, a pointer to the memory where the elements
are stored, data, a pointer to the block owned by the vector,
block, if any, and an ownership flag, owner. The structure
is very simple and looks like this,
typedef struct
{
size_t size;
size_t stride;
double * data;
gsl_block * block;
int owner;
} gsl_vector;
The size is simply the number of vector elements. The range of
valid indices runs from 0 to size-1. The stride is the
step-size from one element to the next in physical memory, measured in
units of the appropriate datatype. The pointer data gives the
location of the first element of the vector in memory. The pointer
block stores the location of the memory block in which the vector
elements are located (if any). If the vector owns this block then the
owner field is set to one and the block will be deallocated when the
vector is freed. If the vector points to a block owned by another
object then the owner field is zero and any underlying block will not be
deallocated with the vector.
The functions for allocating and accessing vectors are defined in gsl_vector.h
Next: Matrices, Previous: Blocks, Up: Vectors and Matrices [Index]
Next: Balancing, Previous: Tridiagonal Systems, Up: Linear Algebra [Index]
These functions calculate the in-place inverse of the triangular matrix T. When
the upper prefix is specified, then the upper triangle of T is used, and when
the lower prefix is specified, the lower triangle is used. If the unit
prefix is specified, then the diagonal elements of the matrix T are taken as
unity and are not referenced. Otherwise the diagonal elements are used in the inversion.
These functions estimate the reciprocal condition number, in the 1-norm, of the upper or lower N-by-N triangular matrix T. The reciprocal condition number is stored in rcond on output, and is defined by 1 / (||T||_1 \cdot ||T^{-1}||_1). Additional workspace of size 3 N is required in work.
Next: Log Complementary Error Function, Previous: Error Function, Up: Error Functions [Index]
These routines compute the complementary error function erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2).
Next: Example programs for B-splines, Previous: Evaluation of B-spline basis function derivatives, Up: Basis Splines [Index]
The Greville abscissae are defined to be the mean location of k-1
consecutive knots in the knot vector for each basis spline function of order
k. With the first and last knots in the gsl_bspline_workspace
knot vector excluded, there are gsl_bspline_ncoeffs Greville abscissae
for any given B-spline basis. These values are often used in B-spline
collocation applications and may also be called Marsden-Schoenberg points.
Returns the location of the i-th Greville abscissa for the given B-spline basis. For the ill-defined case when k=1, the implementation chooses to return breakpoint interval midpoints.
Next: Random Number Generator Performance, Previous: Unix random number generators, Up: Random Number Generation [Index]
The generators in this section are provided for compatibility with existing libraries. If you are converting an existing program to use GSL then you can select these generators to check your new implementation against the original one, using the same random number generator. After verifying that your new program reproduces the original results you can then switch to a higher-quality generator.
Note that most of the generators in this section are based on single linear congruence relations, which are the least sophisticated type of generator. In particular, linear congruences have poor properties when used with a non-prime modulus, as several of these routines do (e.g. with a power of two modulus, 2^31 or 2^32). This leads to periodicity in the least significant bits of each number, with only the higher bits having any randomness. Thus if you want to produce a random bitstream it is best to avoid using the least significant bits.
This is the CRAY random number generator RANF. Its sequence is
x_{n+1} = (a x_n) mod m
defined on 48-bit unsigned integers with a = 44485709377909 and m = 2^48. The seed specifies the lower 32 bits of the initial value, x_1, with the lowest bit set to prevent the seed taking an even value. The upper 16 bits of x_1 are set to 0. A consequence of this procedure is that the pairs of seeds 2 and 3, 4 and 5, etc. produce the same sequences.
The generator compatible with the CRAY MATHLIB routine RANF. It produces double precision floating point numbers which should be identical to those from the original RANF.
There is a subtlety in the implementation of the seeding. The initial state is reversed through one step, by multiplying by the modular inverse of a mod m. This is done for compatibility with the original CRAY implementation.
Note that you can only seed the generator with integers up to 2^32, while the original CRAY implementation uses non-portable wide integers which can cover all 2^48 states of the generator.
The function gsl_rng_get returns the upper 32 bits from each term
of the sequence. The function gsl_rng_uniform uses the full 48
bits to return the double precision number x_n/m.
The period of this generator is 2^46.
This is the RANMAR lagged-fibonacci generator of Marsaglia, Zaman and Tsang. It is a 24-bit generator, originally designed for single-precision IEEE floating point numbers. It was included in the CERNLIB high-energy physics library.
This is the shift-register generator of Kirkpatrick and Stoll. The sequence is based on the recurrence
x_n = x_{n-103} ^^ x_{n-250}
where ^^ denotes “exclusive-or”, defined on 32-bit words. The period of this generator is about 2^250 and it uses 250 words of state per generator.
For more information see,
This is an earlier version of the twisted generalized feedback shift-register generator, and has been superseded by the development of MT19937. However, it is still an acceptable generator in its own right. It has a period of 2^800 and uses 33 words of storage per generator.
For more information see,
This is the VAX generator MTH$RANDOM. Its sequence is,
x_{n+1} = (a x_n + c) mod m
with a = 69069, c = 1 and m = 2^32. The seed specifies the initial value, x_1. The period of this generator is 2^32 and it uses 1 word of storage per generator.
This is the random number generator from the INMOS Transputer Development system. Its sequence is,
x_{n+1} = (a x_n) mod m
with a = 1664525 and m = 2^32. The seed specifies the initial value, x_1.
This is the IBM RANDU generator. Its sequence is
x_{n+1} = (a x_n) mod m
with a = 65539 and m = 2^31. The seed specifies the initial value, x_1. The period of this generator was only 2^29. It has become a textbook example of a poor generator.
This is Park and Miller’s “minimal standard” MINSTD generator, a simple linear congruence which takes care to avoid the major pitfalls of such algorithms. Its sequence is,
x_{n+1} = (a x_n) mod m
with a = 16807 and m = 2^31 - 1 = 2147483647. The seed specifies the initial value, x_1. The period of this generator is about 2^31.
This generator was used in the IMSL Library (subroutine RNUN) and in MATLAB (the RAND function) in the past. It is also sometimes known by the acronym “GGL” (I’m not sure what that stands for).
For more information see,
This is a reimplementation of the 16-bit SLATEC random number generator
RUNIF. A generalization of the generator to 32 bits is provided by
gsl_rng_uni32. The original source code is available from NETLIB.
This is the SLATEC random number generator RAND. It is ancient. The original source code is available from NETLIB.
This is the ZUFALL lagged Fibonacci series generator of Peterson. Its sequence is,
t = u_{n-273} + u_{n-607}
u_n = t - floor(t)
The original source code is available from NETLIB. For more information see,
This is a second-order multiple recursive generator described by Knuth in Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,
x_n = (a_1 x_{n-1} + a_2 x_{n-2}) mod m
with a_1 = 271828183, a_2 = 314159269, and m = 2^31 - 1.
This is a second-order multiple recursive generator described by Knuth
in Seminumerical Algorithms, 3rd Ed., Section 3.6. Knuth
provides its C code. The updated routine gsl_rng_knuthran2002
is from the revised 9th printing and corrects some weaknesses in the
earlier version, which is implemented as gsl_rng_knuthran.
These multiplicative generators are taken from Knuth’s Seminumerical Algorithms, 3rd Ed., pages 106–108. Their sequence is,
x_{n+1} = (a x_n) mod m
where the seed specifies the initial value, x_1. The parameters a and m are as follows, Borosh-Niederreiter: a = 1812433253, m = 2^32, Fishman18: a = 62089911, m = 2^31 - 1, Fishman20: a = 48271, m = 2^31 - 1, L’Ecuyer: a = 40692, m = 2^31 - 249, Waterman: a = 1566083941, m = 2^32.
This is the L’Ecuyer–Fishman random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,
z_{n+1} = (x_n - y_n) mod m
with m = 2^31 - 1.
x_n and y_n are given by the fishman20
and lecuyer21 algorithms.
The seed specifies the initial value,
x_1.
This is the Coveyou random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., Section 3.2.2. Its sequence is,
x_{n+1} = (x_n (x_n + 1)) mod m
with m = 2^32. The seed specifies the initial value, x_1.
Next: Random Number Generator Performance, Previous: Unix random number generators, Up: Random Number Generation [Index]
Next: Complete Orthogonal Decomposition, Previous: QR Decomposition, Up: Linear Algebra [Index]
The QR decomposition of an M-by-N matrix A can be extended to the rank deficient case by introducing a column permutation P,
A P = Q R
The first r columns of Q form an orthonormal basis for the range of A for a matrix with column rank r. This decomposition can also be used to convert the linear system A x = b into the triangular system R y = Q^T b, x = P y, which can be solved by back-substitution and permutation. We denote the QR decomposition with column pivoting by QRP^T since A = Q R P^T. When A is rank deficient with r = {\rm rank}(A), the matrix R can be partitioned as
R = [ R11 R12; 0 R22 ] =~ [ R11 R12; 0 0 ]
where R_{11} is r-by-r and nonsingular. In this case, a “basic” least squares solution for the overdetermined system A x = b can be obtained as
x = P [ R11^-1 c1 ; 0 ]
where c_1 consists of the first r elements of Q^T b. This basic solution is not guaranteed to be the minimum norm solution unless R_{12} = 0 (see Complete Orthogonal Decomposition).
This function factorizes the M-by-N matrix A into the QRP^T decomposition A = Q R P^T. On output the diagonal and upper triangular part of the input matrix contain the matrix R. The permutation matrix P is stored in the permutation p. The sign of the permutation is given by signum. It has the value (-1)^n, where n is the number of interchanges in the permutation. The vector tau and the columns of the lower triangular part of the matrix A contain the Householder coefficients and vectors which encode the orthogonal matrix Q. The vector tau must be of length k=\min(M,N). The matrix Q is related to these components by, Q = Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i v_i^T and v_i is the Householder vector v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)). This is the same storage scheme as used by LAPACK. The vector norm is a workspace of length N used for column pivoting.
The algorithm used to perform the decomposition is Householder QR with column pivoting (Golub & Van Loan, Matrix Computations, Algorithm 5.4.1).
This function factorizes the matrix A into the decomposition A = Q R P^T without modifying A itself and storing the output in the separate matrices q and r.
This function solves the square system A x = b using the QRP^T
decomposition of A held in (QR, tau, p) which must
have been computed previously by gsl_linalg_QRPT_decomp.
This function solves the square system A x = b in-place using the QRP^T decomposition of A held in (QR,tau,p). On input x should contain the right-hand side b, which is replaced by the solution on output.
This function finds the least squares solution to the overdetermined
system A x = b where the matrix A has more rows than
columns and is assumed to have full rank. The least squares solution minimizes
the Euclidean norm of the residual, ||b - A x||. The routine requires as input
the QR decomposition of A into (QR, tau, p) given by
gsl_linalg_QRPT_decomp. The solution is returned in x. The
residual is computed as a by-product and stored in residual. For rank
deficient matrices, gsl_linalg_QRPT_lssolve2 should be used instead.
This function finds the least squares solution to the overdetermined
system A x = b where the matrix A has more rows than
columns and has rank given by the input rank. If the user does not
know the rank of A, the routine gsl_linalg_QRPT_rank can be
called to estimate it. The least squares solution is
the so-called “basic” solution discussed above and may not be the minimum
norm solution. The routine requires as input
the QR decomposition of A into (QR, tau, p) given by
gsl_linalg_QRPT_decomp. The solution is returned in x. The
residual is computed as a by-product and stored in residual.
This function solves the square system R P^T x = Q^T b for x. It can be used when the QR decomposition of a matrix is available in unpacked form as (Q, R).
This function performs a rank-1 update w v^T of the QRP^T decomposition (Q, R, p). The update is given by Q'R' = Q (R + w v^T P) where the output matrices Q' and R' are also orthogonal and right triangular. Note that w is destroyed by the update. The permutation p is not changed.
This function solves the triangular system R P^T x = b for the N-by-N matrix R contained in QR.
This function solves the triangular system R P^T x = b in-place for the N-by-N matrix R contained in QR. On input x should contain the right-hand side b, which is replaced by the solution on output.
This function estimates the rank of the triangular matrix R contained in QR. The algorithm simply counts the number of diagonal elements of R whose absolute value is greater than the specified tolerance tol. If the input tol is negative, a default value of 20 (M + N) eps(max(|diag(R)|)) is used.
This function estimates the reciprocal condition number (using the 1-norm) of the R factor, stored in the upper triangle of QR. The reciprocal condition number estimate, defined as 1 / (||R||_1 \cdot ||R^{-1}||_1), is stored in rcond. Additional workspace of size 3 N is required in work.
Next: Complete Orthogonal Decomposition, Previous: QR Decomposition, Up: Linear Algebra [Index]
Next: Chebyshev Approximation Examples, Previous: Chebyshev Series Evaluation, Up: Chebyshev Approximations [Index]
The following functions allow a Chebyshev series to be differentiated or integrated, producing a new Chebyshev series. Note that the error estimate produced by evaluating the derivative series will be underestimated due to the contribution of higher order terms being neglected.
This function computes the derivative of the series cs, storing the derivative coefficients in the previously allocated deriv. The two series cs and deriv must have been allocated with the same order.
This function computes the integral of the series cs, storing the integral coefficients in the previously allocated integ. The two series cs and integ must have been allocated with the same order. The lower limit of the integration is taken to be the left hand end of the range a.
Next: 1D Interpolation Functions, Up: Interpolation [Index]
Given a set of data points (x_1, y_1) \dots (x_n, y_n) the routines described in this section compute a continuous interpolating function y(x) such that y(x_i) = y_i. The interpolation is piecewise smooth, and its behavior at the end-points is determined by the type of interpolation used.
gsl-ref-html-2.3/LU-Decomposition.html 0000664 0001750 0001750 00000027534 13055414463 015773 0 ustar edd eddNext: QR Decomposition, Up: Linear Algebra [Index]
A general N-by-N square matrix A has an LU decomposition into upper and lower triangular matrices,
P A = L U
where P is a permutation matrix, L is unit lower triangular matrix and U is upper triangular matrix. For square matrices this decomposition can be used to convert the linear system A x = b into a pair of triangular systems (L y = P b, U x = y), which can be solved by forward and back-substitution. Note that the LU decomposition is valid for singular matrices.
These functions factorize the square matrix A into the LU decomposition PA = LU. On output the diagonal and upper triangular part of the input matrix A contain the matrix U. The lower triangular part of the input matrix (excluding the diagonal) contains L. The diagonal elements of L are unity, and are not stored.
The permutation matrix P is encoded in the permutation p on output. The j-th column of the matrix P is given by the k-th column of the identity matrix, where k = p_j the j-th element of the permutation vector. The sign of the permutation is given by signum. It has the value (-1)^n, where n is the number of interchanges in the permutation.
The algorithm used in the decomposition is Gaussian Elimination with partial pivoting (Golub & Van Loan, Matrix Computations, Algorithm 3.4.1).
These functions solve the square system A x = b using the LU
decomposition of A into (LU, p) given by
gsl_linalg_LU_decomp or gsl_linalg_complex_LU_decomp as input.
These functions solve the square system A x = b in-place using the precomputed LU decomposition of A into (LU,p). On input x should contain the right-hand side b, which is replaced by the solution on output.
These functions apply an iterative improvement to x, the solution of A x = b, from the precomputed LU decomposition of A into (LU,p). Additional workspace of length N is required in work.
These functions compute the inverse of a matrix A from its LU decomposition (LU,p), storing the result in the matrix inverse. The inverse is computed by solving the system A x = b for each column of the identity matrix. It is preferable to avoid direct use of the inverse whenever possible, as the linear solver functions can obtain the same result more efficiently and reliably (consult any introductory textbook on numerical linear algebra for details).
These functions compute the determinant of a matrix A from its LU decomposition, LU. The determinant is computed as the product of the diagonal elements of U and the sign of the row permutation signum.
These functions compute the logarithm of the absolute value of the determinant of a matrix A, \ln|\det(A)|, from its LU decomposition, LU. This function may be useful if the direct computation of the determinant would overflow or underflow.
These functions compute the sign or phase factor of the determinant of a matrix A, \det(A)/|\det(A)|, from its LU decomposition, LU.
Next: QR Decomposition, Up: Linear Algebra [Index]
Next: Concept Index, Previous: Variable Index, Up: Top [Index]
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Next: Concept Index, Previous: Variable Index, Up: Top [Index]
Next: Search Bounds and Guesses, Previous: Initializing the Solver, Up: One dimensional Root-Finding [Index]
You must provide a continuous function of one variable for the root finders to operate on, and, sometimes, its first derivative. In order to allow for general parameters the functions are defined by the following data types:
This data type defines a general function with parameters.
double (* function) (double x, void * params)this function should return the value f(x,params) for argument x and parameters params
void * paramsa pointer to the parameters of the function
Here is an example for the general quadratic function,
f(x) = a x^2 + b x + c
with a = 3, b = 2, c = 1. The following code
defines a gsl_function F which you could pass to a root
finder as a function pointer:
struct my_f_params { double a; double b; double c; };
double
my_f (double x, void * p) {
struct my_f_params * params
= (struct my_f_params *)p;
double a = (params->a);
double b = (params->b);
double c = (params->c);
return (a * x + b) * x + c;
}
gsl_function F;
struct my_f_params params = { 3.0, 2.0, 1.0 };
F.function = &my_f;
F.params = ¶ms;
The function f(x) can be evaluated using the macro
GSL_FN_EVAL(&F,x) defined in gsl_math.h.
This data type defines a general function with parameters and its first derivative.
double (* f) (double x, void * params)this function should return the value of f(x,params) for argument x and parameters params
double (* df) (double x, void * params)this function should return the value of the derivative of f with respect to x, f'(x,params), for argument x and parameters params
void (* fdf) (double x, void * params, double * f, double * df)this function should set the values of the function f to f(x,params) and its derivative df to f'(x,params) for argument x and parameters params. This function provides an optimization of the separate functions for f(x) and f'(x)—it is always faster to compute the function and its derivative at the same time.
void * paramsa pointer to the parameters of the function
Here is an example where f(x) = 2\exp(2x):
double
my_f (double x, void * params)
{
return exp (2 * x);
}
double
my_df (double x, void * params)
{
return 2 * exp (2 * x);
}
void
my_fdf (double x, void * params,
double * f, double * df)
{
double t = exp (2 * x);
*f = t;
*df = 2 * t; /* uses existing value */
}
gsl_function_fdf FDF;
FDF.f = &my_f;
FDF.df = &my_df;
FDF.fdf = &my_fdf;
FDF.params = 0;
The function f(x) can be evaluated using the macro
GSL_FN_FDF_EVAL_F(&FDF,x) and the derivative f'(x) can
be evaluated using the macro GSL_FN_FDF_EVAL_DF(&FDF,x). Both
the function y = f(x) and its derivative dy = f'(x) can
be evaluated at the same time using the macro
GSL_FN_FDF_EVAL_F_DF(&FDF,x,y,dy). The macro stores
f(x) in its y argument and f'(x) in its dy
argument—both of these should be pointers to double.
Next: Search Bounds and Guesses, Previous: Initializing the Solver, Up: One dimensional Root-Finding [Index]
Next: 2D Histogram Operations, Previous: Searching 2D histogram ranges, Up: Histograms [Index]
This function returns the maximum value contained in the histogram bins.
This function finds the indices of the bin containing the maximum value in the histogram h and stores the result in (i,j). In the case where several bins contain the same maximum value the first bin found is returned.
This function returns the minimum value contained in the histogram bins.
This function finds the indices of the bin containing the minimum value in the histogram h and stores the result in (i,j). In the case where several bins contain the same maximum value the first bin found is returned.
This function returns the mean of the histogrammed x variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation.
This function returns the mean of the histogrammed y variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation.
This function returns the standard deviation of the histogrammed x variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation.
This function returns the standard deviation of the histogrammed y variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation.
This function returns the covariance of the histogrammed x and y variables, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation.
This function returns the sum of all bin values. Negative bin values are included in the sum.
Next: 2D Histogram Operations, Previous: Searching 2D histogram ranges, Up: Histograms [Index]
Next: Elementary Functions, Previous: Mathematical Constants, Up: Mathematical Functions [Index]
This macro contains the IEEE representation of positive infinity,
+\infty. It is computed from the expression +1.0/0.0.
This macro contains the IEEE representation of negative infinity,
-\infty. It is computed from the expression -1.0/0.0.
This macro contains the IEEE representation of the Not-a-Number symbol,
NaN. It is computed from the ratio 0.0/0.0.
This function returns +1 if x is positive infinity, -1 if x is negative infinity and 0 otherwise.6
This function returns 1 if x is a real number, and 0 if it is infinite or not-a-number.
Note that the C99 standard only requires the
system isinf function to return a non-zero value, without the
sign of the infinity. The implementation in some earlier versions of
GSL used the system isinf function and may have this behavior
on some platforms. Therefore, it is advisable to test the sign of
x separately, if needed, rather than relying the sign of the
return value from gsl_isinf().
Next: Permutation References and Further Reading, Previous: Permutations in cyclic form, Up: Permutations [Index]
The example program below creates a random permutation (by shuffling the elements of the identity) and finds its inverse.
#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_permutation.h>
int
main (void)
{
const size_t N = 10;
const gsl_rng_type * T;
gsl_rng * r;
gsl_permutation * p = gsl_permutation_alloc (N);
gsl_permutation * q = gsl_permutation_alloc (N);
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
printf ("initial permutation:");
gsl_permutation_init (p);
gsl_permutation_fprintf (stdout, p, " %u");
printf ("\n");
printf (" random permutation:");
gsl_ran_shuffle (r, p->data, N, sizeof(size_t));
gsl_permutation_fprintf (stdout, p, " %u");
printf ("\n");
printf ("inverse permutation:");
gsl_permutation_inverse (q, p);
gsl_permutation_fprintf (stdout, q, " %u");
printf ("\n");
gsl_permutation_free (p);
gsl_permutation_free (q);
gsl_rng_free (r);
return 0;
}
Here is the output from the program,
$ ./a.out initial permutation: 0 1 2 3 4 5 6 7 8 9 random permutation: 1 3 5 2 7 6 0 4 9 8 inverse permutation: 6 0 3 1 7 2 5 4 9 8
The random permutation p[i] and its inverse q[i] are
related through the identity p[q[i]] = i, which can be verified
from the output.
The next example program steps forwards through all possible third order permutations, starting from the identity,
#include <stdio.h>
#include <gsl/gsl_permutation.h>
int
main (void)
{
gsl_permutation * p = gsl_permutation_alloc (3);
gsl_permutation_init (p);
do
{
gsl_permutation_fprintf (stdout, p, " %u");
printf ("\n");
}
while (gsl_permutation_next(p) == GSL_SUCCESS);
gsl_permutation_free (p);
return 0;
}
Here is the output from the program,
$ ./a.out 0 1 2 0 2 1 1 0 2 1 2 0 2 0 1 2 1 0
The permutations are generated in lexicographic order. To reverse the
sequence, begin with the final permutation (which is the reverse of the
identity) and replace gsl_permutation_next with
gsl_permutation_prev.
Next: Permutation References and Further Reading, Previous: Permutations in cyclic form, Up: Permutations [Index]
Next: Compatibility with C++, Previous: Alternative optimized functions, Up: Using the library [Index]
Many functions in the library are defined for different numeric types.
This feature is implemented by varying the name of the function with a
type-related modifier—a primitive form of C++ templates. The
modifier is inserted into the function name after the initial module
prefix. The following table shows the function names defined for all
the numeric types of an imaginary module gsl_foo with function
fn,
gsl_foo_fn double gsl_foo_long_double_fn long double gsl_foo_float_fn float gsl_foo_long_fn long gsl_foo_ulong_fn unsigned long gsl_foo_int_fn int gsl_foo_uint_fn unsigned int gsl_foo_short_fn short gsl_foo_ushort_fn unsigned short gsl_foo_char_fn char gsl_foo_uchar_fn unsigned char
The normal numeric precision double is considered the default and
does not require a suffix. For example, the function
gsl_stats_mean computes the mean of double precision numbers,
while the function gsl_stats_int_mean computes the mean of
integers.
A corresponding scheme is used for library defined types, such as
gsl_vector and gsl_matrix. In this case the modifier is
appended to the type name. For example, if a module defines a new
type-dependent struct or typedef gsl_foo it is modified for other
types in the following way,
gsl_foo double gsl_foo_long_double long double gsl_foo_float float gsl_foo_long long gsl_foo_ulong unsigned long gsl_foo_int int gsl_foo_uint unsigned int gsl_foo_short short gsl_foo_ushort unsigned short gsl_foo_char char gsl_foo_uchar unsigned char
When a module contains type-dependent definitions the library provides individual header files for each type. The filenames are modified as shown in the below. For convenience the default header includes the definitions for all the types. To include only the double precision header file, or any other specific type, use its individual filename.
#include <gsl/gsl_foo.h> All types #include <gsl/gsl_foo_double.h> double #include <gsl/gsl_foo_long_double.h> long double #include <gsl/gsl_foo_float.h> float #include <gsl/gsl_foo_long.h> long #include <gsl/gsl_foo_ulong.h> unsigned long #include <gsl/gsl_foo_int.h> int #include <gsl/gsl_foo_uint.h> unsigned int #include <gsl/gsl_foo_short.h> short #include <gsl/gsl_foo_ushort.h> unsigned short #include <gsl/gsl_foo_char.h> char #include <gsl/gsl_foo_uchar.h> unsigned char
Next: Compatibility with C++, Previous: Alternative optimized functions, Up: Using the library [Index]
Next: 2D Histogram Statistics, Previous: Updating and accessing 2D histogram elements, Up: Histograms [Index]
The following functions are used by the access and update routines to locate the bin which corresponds to a given (x,y) coordinate.
This function finds and sets the indices i and j to
the bin which covers the coordinates (x,y). The bin is
located using a binary search. The search includes an optimization for
histograms with uniform ranges, and will return the correct bin immediately
in this case. If (x,y) is found then the function sets the
indices (i,j) and returns GSL_SUCCESS. If
(x,y) lies outside the valid range of the histogram then the
function returns GSL_EDOM and the error handler is invoked.
Previous: Root Finding Examples, Up: One dimensional Root-Finding [Index]
For information on the Brent-Dekker algorithm see the following two papers,
Next: Irregular Bessel Functions - Fractional Order, Previous: Irregular Modified Spherical Bessel Functions, Up: Bessel Functions [Index]
These routines compute the regular cylindrical Bessel function of fractional order \nu, J_\nu(x).
This function computes the regular cylindrical Bessel function of fractional order \nu, J_\nu(x), evaluated at a series of x values. The array v of length size contains the x values. They are assumed to be strictly ordered and positive. The array is over-written with the values of J_\nu(x_i).
Next: Contributors to GSL, Previous: IEEE floating-point arithmetic, Up: Top [Index]
This chapter describes some tips and tricks for debugging numerical programs which use GSL.
| • Using gdb: | ||
| • Examining floating point registers: | ||
| • Handling floating point exceptions: | ||
| • GCC warning options for numerical programs: | ||
| • Debugging References: |
Next: Root Finding References and Further Reading, Previous: Root Finding Algorithms using Derivatives, Up: One dimensional Root-Finding [Index]
For any root finding algorithm we need to prepare the function to be solved. For this example we will use the general quadratic equation described earlier. We first need a header file (demo_fn.h) to define the function parameters,
struct quadratic_params
{
double a, b, c;
};
double quadratic (double x, void *params);
double quadratic_deriv (double x, void *params);
void quadratic_fdf (double x, void *params,
double *y, double *dy);
We place the function definitions in a separate file (demo_fn.c),
double
quadratic (double x, void *params)
{
struct quadratic_params *p
= (struct quadratic_params *) params;
double a = p->a;
double b = p->b;
double c = p->c;
return (a * x + b) * x + c;
}
double
quadratic_deriv (double x, void *params)
{
struct quadratic_params *p
= (struct quadratic_params *) params;
double a = p->a;
double b = p->b;
return 2.0 * a * x + b;
}
void
quadratic_fdf (double x, void *params,
double *y, double *dy)
{
struct quadratic_params *p
= (struct quadratic_params *) params;
double a = p->a;
double b = p->b;
double c = p->c;
*y = (a * x + b) * x + c;
*dy = 2.0 * a * x + b;
}
The first program uses the function solver gsl_root_fsolver_brent
for Brent’s method and the general quadratic defined above to solve the
following equation,
x^2 - 5 = 0
with solution x = \sqrt 5 = 2.236068...
#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_roots.h>
#include "demo_fn.h"
#include "demo_fn.c"
int
main (void)
{
int status;
int iter = 0, max_iter = 100;
const gsl_root_fsolver_type *T;
gsl_root_fsolver *s;
double r = 0, r_expected = sqrt (5.0);
double x_lo = 0.0, x_hi = 5.0;
gsl_function F;
struct quadratic_params params = {1.0, 0.0, -5.0};
F.function = &quadratic;
F.params = ¶ms;
T = gsl_root_fsolver_brent;
s = gsl_root_fsolver_alloc (T);
gsl_root_fsolver_set (s, &F, x_lo, x_hi);
printf ("using %s method\n",
gsl_root_fsolver_name (s));
printf ("%5s [%9s, %9s] %9s %10s %9s\n",
"iter", "lower", "upper", "root",
"err", "err(est)");
do
{
iter++;
status = gsl_root_fsolver_iterate (s);
r = gsl_root_fsolver_root (s);
x_lo = gsl_root_fsolver_x_lower (s);
x_hi = gsl_root_fsolver_x_upper (s);
status = gsl_root_test_interval (x_lo, x_hi,
0, 0.001);
if (status == GSL_SUCCESS)
printf ("Converged:\n");
printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
iter, x_lo, x_hi,
r, r - r_expected,
x_hi - x_lo);
}
while (status == GSL_CONTINUE && iter < max_iter);
gsl_root_fsolver_free (s);
return status;
}
Here are the results of the iterations,
$ ./a.out
using brent method
iter [ lower, upper] root err err(est)
1 [1.0000000, 5.0000000] 1.0000000 -1.2360680 4.0000000
2 [1.0000000, 3.0000000] 3.0000000 +0.7639320 2.0000000
3 [2.0000000, 3.0000000] 2.0000000 -0.2360680 1.0000000
4 [2.2000000, 3.0000000] 2.2000000 -0.0360680 0.8000000
5 [2.2000000, 2.2366300] 2.2366300 +0.0005621 0.0366300
Converged:
6 [2.2360634, 2.2366300] 2.2360634 -0.0000046 0.0005666
If the program is modified to use the bisection solver instead of
Brent’s method, by changing gsl_root_fsolver_brent to
gsl_root_fsolver_bisection the slower convergence of the
Bisection method can be observed,
$ ./a.out
using bisection method
iter [ lower, upper] root err err(est)
1 [0.0000000, 2.5000000] 1.2500000 -0.9860680 2.5000000
2 [1.2500000, 2.5000000] 1.8750000 -0.3610680 1.2500000
3 [1.8750000, 2.5000000] 2.1875000 -0.0485680 0.6250000
4 [2.1875000, 2.5000000] 2.3437500 +0.1076820 0.3125000
5 [2.1875000, 2.3437500] 2.2656250 +0.0295570 0.1562500
6 [2.1875000, 2.2656250] 2.2265625 -0.0095055 0.0781250
7 [2.2265625, 2.2656250] 2.2460938 +0.0100258 0.0390625
8 [2.2265625, 2.2460938] 2.2363281 +0.0002601 0.0195312
9 [2.2265625, 2.2363281] 2.2314453 -0.0046227 0.0097656
10 [2.2314453, 2.2363281] 2.2338867 -0.0021813 0.0048828
11 [2.2338867, 2.2363281] 2.2351074 -0.0009606 0.0024414
Converged:
12 [2.2351074, 2.2363281] 2.2357178 -0.0003502 0.0012207
The next program solves the same function using a derivative solver instead.
#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_roots.h>
#include "demo_fn.h"
#include "demo_fn.c"
int
main (void)
{
int status;
int iter = 0, max_iter = 100;
const gsl_root_fdfsolver_type *T;
gsl_root_fdfsolver *s;
double x0, x = 5.0, r_expected = sqrt (5.0);
gsl_function_fdf FDF;
struct quadratic_params params = {1.0, 0.0, -5.0};
FDF.f = &quadratic;
FDF.df = &quadratic_deriv;
FDF.fdf = &quadratic_fdf;
FDF.params = ¶ms;
T = gsl_root_fdfsolver_newton;
s = gsl_root_fdfsolver_alloc (T);
gsl_root_fdfsolver_set (s, &FDF, x);
printf ("using %s method\n",
gsl_root_fdfsolver_name (s));
printf ("%-5s %10s %10s %10s\n",
"iter", "root", "err", "err(est)");
do
{
iter++;
status = gsl_root_fdfsolver_iterate (s);
x0 = x;
x = gsl_root_fdfsolver_root (s);
status = gsl_root_test_delta (x, x0, 0, 1e-3);
if (status == GSL_SUCCESS)
printf ("Converged:\n");
printf ("%5d %10.7f %+10.7f %10.7f\n",
iter, x, x - r_expected, x - x0);
}
while (status == GSL_CONTINUE && iter < max_iter);
gsl_root_fdfsolver_free (s);
return status;
}
Here are the results for Newton’s method,
$ ./a.out
using newton method
iter root err err(est)
1 3.0000000 +0.7639320 -2.0000000
2 2.3333333 +0.0972654 -0.6666667
3 2.2380952 +0.0020273 -0.0952381
Converged:
4 2.2360689 +0.0000009 -0.0020263
Note that the error can be estimated more accurately by taking the
difference between the current iterate and next iterate rather than the
previous iterate. The other derivative solvers can be investigated by
changing gsl_root_fdfsolver_newton to
gsl_root_fdfsolver_secant or
gsl_root_fdfsolver_steffenson.
Next: Root Finding References and Further Reading, Previous: Root Finding Algorithms using Derivatives, Up: One dimensional Root-Finding [Index]
Previous: Linear Algebra Examples, Up: Linear Algebra [Index]
Further information on the algorithms described in this section can be found in the following book,
The LAPACK library is described in the following manual,
The LAPACK source code can be found at the website above, along with an online copy of the users guide.
The Modified Golub-Reinsch algorithm is described in the following paper,
The Jacobi algorithm for singular value decomposition is described in the following papers,
lawns or
lawnspdf directories.
The algorithm for estimating a matrix condition number is described in the following paper,
Previous: Linear Algebra Examples, Up: Linear Algebra [Index]
Next: Sparse Linear Algebra, Previous: Sparse Matrices, Up: Top [Index]
The Sparse Basic Linear Algebra Subprograms (BLAS) define a set of fundamental operations on vectors and sparse matrices which can be used to create optimized higher-level linear algebra functionality. GSL supports a limited number of BLAS operations for sparse matrices.
The header file gsl_spblas.h contains the prototypes for the sparse BLAS functions and related declarations.
| • Sparse BLAS operations: | ||
| • Sparse BLAS References and Further Reading: |
Next: Covariance, Previous: Higher moments (skewness and kurtosis), Up: Statistics [Index]
This function computes the lag-1 autocorrelation of the dataset data.
a_1 = {\sum_{i = 2}^{n} (x_{i} - \Hat\mu) (x_{i-1} - \Hat\mu)
\over
\sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i} - \Hat\mu)}
This function computes the lag-1 autocorrelation of the dataset data using the given value of the mean mean.
Previous: Coulomb Wave Functions, Up: Coulomb Functions [Index]
The Coulomb wave function normalization constant is defined in Abramowitz 14.1.7.
This function computes the Coulomb wave function normalization constant C_L(\eta) for L > -1.
This function computes the Coulomb wave function normalization constant C_L(\eta) for L = Lmin \dots Lmin + kmax, Lmin > -1.
Previous: Numerical Differentiation Examples, Up: Numerical Differentiation [Index]
The algorithms used by these functions are described in the following sources:
Next: Adaptive Step-size Control, Previous: Defining the ODE System, Up: Ordinary Differential Equations [Index]
The lowest level components are the stepping functions which advance a solution from time t to t+h for a fixed step-size h and estimate the resulting local error.
This function returns a pointer to a newly allocated instance of a stepping function of type T for a system of dim dimensions. Please note that if you use a stepper method that requires access to a driver object, it is advisable to use a driver allocation method, which automatically allocates a stepper, too.
This function resets the stepping function s. It should be used whenever the next use of s will not be a continuation of a previous step.
This function frees all the memory associated with the stepping function s.
This function returns a pointer to the name of the stepping function. For example,
printf ("step method is '%s'\n",
gsl_odeiv2_step_name (s));
would print something like step method is 'rkf45'.
This function returns the order of the stepping function on the previous step. The order can vary if the stepping function itself is adaptive.
This function sets a pointer of the driver object d for stepper s, to allow the stepper to access control (and evolve) object through the driver object. This is a requirement for some steppers, to get the desired error level for internal iteration of stepper. Allocation of a driver object calls this function automatically.
This function applies the stepping function s to the system of equations defined by sys, using the step-size h to advance the system from time t and state y to time t+h. The new state of the system is stored in y on output, with an estimate of the absolute error in each component stored in yerr. If the argument dydt_in is not null it should point an array containing the derivatives for the system at time t on input. This is optional as the derivatives will be computed internally if they are not provided, but allows the reuse of existing derivative information. On output the new derivatives of the system at time t+h will be stored in dydt_out if it is not null.
The stepping function returns GSL_FAILURE if it is unable to
compute the requested step. Also, if the user-supplied functions
defined in the system sys return a status other than
GSL_SUCCESS the step will be aborted. In that case, the
elements of y will be restored to their pre-step values and the
error code from the user-supplied function will be returned. Failure
may be due to a singularity in the system or too large step-size
h. In that case the step should be attempted again with a
smaller step-size, e.g. h/2.
If the driver object is not appropriately set via
gsl_odeiv2_step_set_driver for those steppers that need it, the
stepping function returns GSL_EFAULT. If the user-supplied
functions defined in the system sys returns GSL_EBADFUNC,
the function returns immediately with the same return code. In this
case the user must call gsl_odeiv2_step_reset before calling
this function again.
The following algorithms are available,
Explicit 4th order (classical) Runge-Kutta. Error estimation is carried out by the step doubling method. For more efficient estimate of the error, use the embedded methods described below.
Explicit embedded Runge-Kutta-Fehlberg (4, 5) method. This method is a good general-purpose integrator.
Implicit Gaussian first order Runge-Kutta. Also known as implicit
Euler or backward Euler method. Error estimation is carried out by the
step doubling method. This algorithm requires the Jacobian and
access to the driver object via gsl_odeiv2_step_set_driver.
Implicit Gaussian second order Runge-Kutta. Also known as implicit
mid-point rule. Error estimation is carried out by the step doubling
method. This stepper requires the Jacobian and access to the driver
object via gsl_odeiv2_step_set_driver.
Implicit Gaussian 4th order Runge-Kutta. Error estimation is carried
out by the step doubling method. This algorithm requires the Jacobian
and access to the driver object via gsl_odeiv2_step_set_driver.
Implicit Bulirsch-Stoer method of Bader and Deuflhard. The method is generally suitable for stiff problems. This stepper requires the Jacobian.
A variable-coefficient linear multistep Adams method in Nordsieck
form. This stepper uses explicit Adams-Bashforth (predictor) and
implicit Adams-Moulton (corrector) methods in P(EC)^m
functional iteration mode. Method order varies dynamically between 1
and 12. This stepper requires the access to the driver object via
gsl_odeiv2_step_set_driver.
A variable-coefficient linear multistep backward differentiation
formula (BDF) method in Nordsieck form. This stepper uses the explicit
BDF formula as predictor and implicit BDF formula as corrector. A
modified Newton iteration method is used to solve the system of
non-linear equations. Method order varies dynamically between 1 and
5. The method is generally suitable for stiff problems. This stepper
requires the Jacobian and the access to the driver object via
gsl_odeiv2_step_set_driver.
Next: Adaptive Step-size Control, Previous: Defining the ODE System, Up: Ordinary Differential Equations [Index]
Next: Example programs for 2D histograms, Previous: Reading and writing 2D histograms, Up: Histograms [Index]
As in the one-dimensional case, a two-dimensional histogram made by counting events can be regarded as a measurement of a probability distribution. Allowing for statistical error, the height of each bin represents the probability of an event where (x,y) falls in the range of that bin. For a two-dimensional histogram the probability distribution takes the form p(x,y) dx dy where,
p(x,y) = n_{ij}/ (N A_{ij})
In this equation n_{ij} is the number of events in the bin which contains (x,y), A_{ij} is the area of the bin and N is the total number of events. The distribution of events within each bin is assumed to be uniform.
size_t nx, nyThis is the number of histogram bins used to approximate the probability distribution function in the x and y directions.
double * xrangeThe ranges of the bins in the x-direction are stored in an array of nx + 1 elements pointed to by xrange.
double * yrangeThe ranges of the bins in the y-direction are stored in an array of ny + 1 pointed to by yrange.
double * sumThe cumulative probability for the bins is stored in an array of nx*ny elements pointed to by sum.
The following functions allow you to create a gsl_histogram2d_pdf
struct which represents a two dimensional probability distribution and
generate random samples from it.
This function allocates memory for a two-dimensional probability
distribution of size nx-by-ny and returns a pointer to a
newly initialized gsl_histogram2d_pdf struct. If insufficient
memory is available a null pointer is returned and the error handler is
invoked with an error code of GSL_ENOMEM.
This function initializes the two-dimensional probability distribution
calculated p from the histogram h. If any of the bins of
h are negative then the error handler is invoked with an error
code of GSL_EDOM because a probability distribution cannot
contain negative values.
This function frees the two-dimensional probability distribution function p and all of the memory associated with it.
This function uses two uniform random numbers between zero and one, r1 and r2, to compute a single random sample from the two-dimensional probability distribution p.
Next: Example programs for 2D histograms, Previous: Reading and writing 2D histograms, Up: Histograms [Index]
Next: Exponential Functions, Previous: Elliptic Functions (Jacobi), Up: Special Functions [Index]
The error function is described in Abramowitz & Stegun, Chapter 7. The functions in this section are declared in the header file gsl_sf_erf.h.
| • Error Function: | ||
| • Complementary Error Function: | ||
| • Log Complementary Error Function: | ||
| • Probability functions: |
Next: Providing the function to solve, Previous: Root Finding Caveats, Up: One dimensional Root-Finding [Index]
This function returns a pointer to a newly allocated instance of a solver of type T. For example, the following code creates an instance of a bisection solver,
const gsl_root_fsolver_type * T = gsl_root_fsolver_bisection; gsl_root_fsolver * s = gsl_root_fsolver_alloc (T);
If there is insufficient memory to create the solver then the function
returns a null pointer and the error handler is invoked with an error
code of GSL_ENOMEM.
This function returns a pointer to a newly allocated instance of a derivative-based solver of type T. For example, the following code creates an instance of a Newton-Raphson solver,
const gsl_root_fdfsolver_type * T = gsl_root_fdfsolver_newton; gsl_root_fdfsolver * s = gsl_root_fdfsolver_alloc (T);
If there is insufficient memory to create the solver then the function
returns a null pointer and the error handler is invoked with an error
code of GSL_ENOMEM.
This function initializes, or reinitializes, an existing solver s to use the function f and the initial search interval [x_lower, x_upper].
This function initializes, or reinitializes, an existing solver s to use the function and derivative fdf and the initial guess root.
These functions free all the memory associated with the solver s.
These functions return a pointer to the name of the solver. For example,
printf ("s is a '%s' solver\n",
gsl_root_fsolver_name (s));
would print something like s is a 'bisection' solver.
Next: Providing the function to solve, Previous: Root Finding Caveats, Up: One dimensional Root-Finding [Index]
Next: Example programs for histograms, Previous: Resampling from histograms, Up: Histograms [Index]
The probability distribution function for a histogram consists of a set of bins which measure the probability of an event falling into a given range of a continuous variable x. A probability distribution function is defined by the following struct, which actually stores the cumulative probability distribution function. This is the natural quantity for generating samples via the inverse transform method, because there is a one-to-one mapping between the cumulative probability distribution and the range [0,1]. It can be shown that by taking a uniform random number in this range and finding its corresponding coordinate in the cumulative probability distribution we obtain samples with the desired probability distribution.
size_t nThis is the number of bins used to approximate the probability distribution function.
double * rangeThe ranges of the bins are stored in an array of n+1 elements pointed to by range.
double * sumThe cumulative probability for the bins is stored in an array of n elements pointed to by sum.
The following functions allow you to create a gsl_histogram_pdf
struct which represents this probability distribution and generate
random samples from it.
This function allocates memory for a probability distribution with
n bins and returns a pointer to a newly initialized
gsl_histogram_pdf struct. If insufficient memory is available a
null pointer is returned and the error handler is invoked with an error
code of GSL_ENOMEM.
This function initializes the probability distribution p with
the contents of the histogram h. If any of the bins of h are
negative then the error handler is invoked with an error code of
GSL_EDOM because a probability distribution cannot contain
negative values.
This function frees the probability distribution function p and all of the memory associated with it.
This function uses r, a uniform random number between zero and one, to compute a single random sample from the probability distribution p. The algorithm used to compute the sample s is given by the following formula,
s = range[i] + delta * (range[i+1] - range[i])
where i is the index which satisfies sum[i] <= r < sum[i+1] and delta is (r - sum[i])/(sum[i+1] - sum[i]).
Next: Example programs for histograms, Previous: Resampling from histograms, Up: Histograms [Index]
Next: The Rayleigh Distribution, Previous: The Exponential Power Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the Cauchy distribution with scale parameter a. The probability distribution for Cauchy random variates is,
p(x) dx = {1 \over a\pi (1 + (x/a)^2) } dx
for x in the range -\infty to +\infty. The Cauchy distribution is also known as the Lorentz distribution.
This function computes the probability density p(x) at x for a Cauchy distribution with scale parameter a, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Cauchy distribution with scale parameter a.
Next: The Landau Distribution, Previous: The Rayleigh Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the tail of the Rayleigh distribution with scale parameter sigma and a lower limit of a. The distribution is,
p(x) dx = {x \over \sigma^2} \exp ((a^2 - x^2) /(2 \sigma^2)) dx
for x > a.
This function computes the probability density p(x) at x for a Rayleigh tail distribution with scale parameter sigma and lower limit a, using the formula given above.
Next: Example programs for matrices, Previous: Finding maximum and minimum elements of matrices, Up: Matrices [Index]
The following functions are defined for real and complex matrices. For complex matrices both the real and imaginary parts must satisfy the conditions.
These functions return 1 if all the elements of the matrix m are zero, strictly positive, strictly negative, or non-negative respectively, and 0 otherwise. To test whether a matrix is positive-definite, use the Cholesky decomposition (see Cholesky Decomposition).
This function returns 1 if the matrices a and b are equal (by comparison of element values) and 0 otherwise.
Next: Evaluation of B-spline basis functions, Previous: Initializing the B-splines solver, Up: Basis Splines [Index]
This function computes the knots associated with the given breakpoints
and stores them internally in w->knots.
This function assumes uniformly spaced breakpoints on [a,b]
and constructs the corresponding knot vector using the previously
specified nbreak parameter. The knots are stored in
w->knots.
Next: Reading and writing blocks, Up: Blocks [Index]
The functions for allocating memory to a block follow the style of
malloc and free. In addition they also perform their own
error checking. If there is insufficient memory available to allocate a
block then the functions call the GSL error handler (with an error
number of GSL_ENOMEM) in addition to returning a null
pointer. Thus if you use the library error handler to abort your program
then it isn’t necessary to check every alloc.
This function allocates memory for a block of n double-precision
elements, returning a pointer to the block struct. The block is not
initialized and so the values of its elements are undefined. Use the
function gsl_block_calloc if you want to ensure that all the
elements are initialized to zero.
A null pointer is returned if insufficient memory is available to create the block.
This function allocates memory for a block and initializes all the elements of the block to zero.
This function frees the memory used by a block b previously
allocated with gsl_block_alloc or gsl_block_calloc.
Next: Auxiliary random number generator functions, Previous: Random number generator initialization, Up: Random Number Generation [Index]
The following functions return uniformly distributed random numbers,
either as integers or double precision floating point numbers. Inline versions of these functions are used when HAVE_INLINE is defined.
To obtain non-uniform distributions see Random Number Distributions.
This function returns a random integer from the generator r. The
minimum and maximum values depend on the algorithm used, but all
integers in the range [min,max] are equally likely. The
values of min and max can be determined using the auxiliary
functions gsl_rng_max (r) and gsl_rng_min (r).
This function returns a double precision floating point number uniformly
distributed in the range [0,1). The range includes 0.0 but excludes 1.0.
The value is typically obtained by dividing the result of
gsl_rng_get(r) by gsl_rng_max(r) + 1.0 in double
precision. Some generators compute this ratio internally so that they
can provide floating point numbers with more than 32 bits of randomness
(the maximum number of bits that can be portably represented in a single
unsigned long int).
This function returns a positive double precision floating point number
uniformly distributed in the range (0,1), excluding both 0.0 and 1.0.
The number is obtained by sampling the generator with the algorithm of
gsl_rng_uniform until a non-zero value is obtained. You can use
this function if you need to avoid a singularity at 0.0.
This function returns a random integer from 0 to n-1 inclusive by scaling down and/or discarding samples from the generator r. All integers in the range [0,n-1] are produced with equal probability. For generators with a non-zero minimum value an offset is applied so that zero is returned with the correct probability.
Note that this function is designed for sampling from ranges smaller
than the range of the underlying generator. The parameter n
must be less than or equal to the range of the generator r.
If n is larger than the range of the generator then the function
calls the error handler with an error code of GSL_EINVAL and
returns zero.
In particular, this function is not intended for generating the full range of
unsigned integer values [0,2^32-1]. Instead
choose a generator with the maximal integer range and zero minimum
value, such as gsl_rng_ranlxd1, gsl_rng_mt19937 or
gsl_rng_taus, and sample it directly using
gsl_rng_get. The range of each generator can be found using
the auxiliary functions described in the next section.
Next: Auxiliary random number generator functions, Previous: Random number generator initialization, Up: Random Number Generation [Index]
Previous: Beta Functions, Up: Gamma and Beta Functions [Index]
These routines compute the normalized incomplete Beta function I_x(a,b)=B_x(a,b)/B(a,b) where B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt for 0 <= x <= 1. For a > 0, b > 0 the value is computed using a continued fraction expansion. For all other values it is computed using the relation I_x(a,b,x) = (1/a) x^a 2F1(a,1-b,a+1,x)/B(a,b).
Next: Level 3 GSL BLAS Interface, Previous: Level 1 GSL BLAS Interface, Up: GSL BLAS Interface [Index]
These functions compute the matrix-vector product and sum y =
\alpha op(A) x + \beta y, where op(A) = A,
A^T, A^H for TransA = CblasNoTrans,
CblasTrans, CblasConjTrans.
These functions compute the matrix-vector product
x = op(A) x for the triangular matrix A, where
op(A) = A, A^T, A^H for TransA =
CblasNoTrans, CblasTrans, CblasConjTrans. When
Uplo is CblasUpper then the upper triangle of A is
used, and when Uplo is CblasLower then the lower triangle
of A is used. If Diag is CblasNonUnit then the
diagonal of the matrix is used, but if Diag is CblasUnit
then the diagonal elements of the matrix A are taken as unity and
are not referenced.
These functions compute inv(op(A)) x for x, where
op(A) = A, A^T, A^H for TransA =
CblasNoTrans, CblasTrans, CblasConjTrans. When
Uplo is CblasUpper then the upper triangle of A is
used, and when Uplo is CblasLower then the lower triangle
of A is used. If Diag is CblasNonUnit then the
diagonal of the matrix is used, but if Diag is CblasUnit
then the diagonal elements of the matrix A are taken as unity and
are not referenced.
These functions compute the matrix-vector product and sum y =
\alpha A x + \beta y for the symmetric matrix A. Since the
matrix A is symmetric only its upper half or lower half need to be
stored. When Uplo is CblasUpper then the upper triangle
and diagonal of A are used, and when Uplo is
CblasLower then the lower triangle and diagonal of A are
used.
These functions compute the matrix-vector product and sum y =
\alpha A x + \beta y for the hermitian matrix A. Since the
matrix A is hermitian only its upper half or lower half need to be
stored. When Uplo is CblasUpper then the upper triangle
and diagonal of A are used, and when Uplo is
CblasLower then the lower triangle and diagonal of A are
used. The imaginary elements of the diagonal are automatically assumed
to be zero and are not referenced.
These functions compute the rank-1 update A = \alpha x y^T + A of the matrix A.
These functions compute the conjugate rank-1 update A = \alpha x y^H + A of the matrix A.
These functions compute the symmetric rank-1 update A = \alpha x
x^T + A of the symmetric matrix A. Since the matrix A is
symmetric only its upper half or lower half need to be stored. When
Uplo is CblasUpper then the upper triangle and diagonal of
A are used, and when Uplo is CblasLower then the
lower triangle and diagonal of A are used.
These functions compute the hermitian rank-1 update A = \alpha x
x^H + A of the hermitian matrix A. Since the matrix A is
hermitian only its upper half or lower half need to be stored. When
Uplo is CblasUpper then the upper triangle and diagonal of
A are used, and when Uplo is CblasLower then the
lower triangle and diagonal of A are used. The imaginary elements
of the diagonal are automatically set to zero.
These functions compute the symmetric rank-2 update A = \alpha x
y^T + \alpha y x^T + A of the symmetric matrix A. Since the
matrix A is symmetric only its upper half or lower half need to be
stored. When Uplo is CblasUpper then the upper triangle
and diagonal of A are used, and when Uplo is
CblasLower then the lower triangle and diagonal of A are
used.
These functions compute the hermitian rank-2 update A = \alpha x
y^H + \alpha^* y x^H + A of the hermitian matrix A. Since the
matrix A is hermitian only its upper half or lower half need to be
stored. When Uplo is CblasUpper then the upper triangle
and diagonal of A are used, and when Uplo is
CblasLower then the lower triangle and diagonal of A are
used. The imaginary elements of the diagonal are automatically set to zero.
Next: Level 3 GSL BLAS Interface, Previous: Level 1 GSL BLAS Interface, Up: GSL BLAS Interface [Index]
Next: FFT References and Further Reading, Previous: Radix-2 FFT routines for real data, Up: Fast Fourier Transforms [Index]
This section describes mixed-radix FFT algorithms for real data. The mixed-radix functions work for FFTs of any length. They are a reimplementation of the real-FFT routines in the Fortran FFTPACK library by Paul Swarztrauber. The theory behind the algorithm is explained in the article Fast Mixed-Radix Real Fourier Transforms by Clive Temperton. The routines here use the same indexing scheme and basic algorithms as FFTPACK.
The functions use the FFTPACK storage convention for half-complex sequences. In this convention the half-complex transform of a real sequence is stored with frequencies in increasing order, starting at zero, with the real and imaginary parts of each frequency in neighboring locations. When a value is known to be real the imaginary part is not stored. The imaginary part of the zero-frequency component is never stored. It is known to be zero (since the zero frequency component is simply the sum of the input data (all real)). For a sequence of even length the imaginary part of the frequency n/2 is not stored either, since the symmetry z_k = z_{n-k}^* implies that this is purely real too.
The storage scheme is best shown by some examples. The table below
shows the output for an odd-length sequence, n=5. The two columns
give the correspondence between the 5 values in the half-complex
sequence returned by gsl_fft_real_transform, halfcomplex[] and the
values complex[] that would be returned if the same real input
sequence were passed to gsl_fft_complex_backward as a complex
sequence (with imaginary parts set to 0),
complex[0].real = halfcomplex[0] complex[0].imag = 0 complex[1].real = halfcomplex[1] complex[1].imag = halfcomplex[2] complex[2].real = halfcomplex[3] complex[2].imag = halfcomplex[4] complex[3].real = halfcomplex[3] complex[3].imag = -halfcomplex[4] complex[4].real = halfcomplex[1] complex[4].imag = -halfcomplex[2]
The upper elements of the complex array, complex[3] and
complex[4] are filled in using the symmetry condition. The
imaginary part of the zero-frequency term complex[0].imag is
known to be zero by the symmetry.
The next table shows the output for an even-length sequence, n=6. In the even case there are two values which are purely real,
complex[0].real = halfcomplex[0] complex[0].imag = 0 complex[1].real = halfcomplex[1] complex[1].imag = halfcomplex[2] complex[2].real = halfcomplex[3] complex[2].imag = halfcomplex[4] complex[3].real = halfcomplex[5] complex[3].imag = 0 complex[4].real = halfcomplex[3] complex[4].imag = -halfcomplex[4] complex[5].real = halfcomplex[1] complex[5].imag = -halfcomplex[2]
The upper elements of the complex array, complex[4] and
complex[5] are filled in using the symmetry condition. Both
complex[0].imag and complex[3].imag are known to be zero.
All these functions are declared in the header files gsl_fft_real.h and gsl_fft_halfcomplex.h.
These functions prepare trigonometric lookup tables for an FFT of size
n real elements. The functions return a pointer to the newly
allocated struct if no errors were detected, and a null pointer in the
case of error. The length n is factorized into a product of
subtransforms, and the factors and their trigonometric coefficients are
stored in the wavetable. The trigonometric coefficients are computed
using direct calls to sin and cos, for accuracy.
Recursion relations could be used to compute the lookup table faster,
but if an application performs many FFTs of the same length then
computing the wavetable is a one-off overhead which does not affect the
final throughput.
The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified by calls to any of the other FFT functions. The appropriate type of wavetable must be used for forward real or inverse half-complex transforms.
These functions free the memory associated with the wavetable wavetable. The wavetable can be freed if no further FFTs of the same length will be needed.
The mixed radix algorithms require additional working space to hold the intermediate steps of the transform,
This function allocates a workspace for a real transform of length n. The same workspace can be used for both forward real and inverse halfcomplex transforms.
This function frees the memory associated with the workspace workspace. The workspace can be freed if no further FFTs of the same length will be needed.
The following functions compute the transforms of real and half-complex data,
These functions compute the FFT of data, a real or half-complex
array of length n, using a mixed radix decimation-in-frequency
algorithm. For gsl_fft_real_transform data is an array of
time-ordered real data. For gsl_fft_halfcomplex_transform
data contains Fourier coefficients in the half-complex ordering
described above. There is no restriction on the length n.
Efficient modules are provided for subtransforms of length 2, 3, 4 and
5. Any remaining factors are computed with a slow, O(n^2),
general-n module. The caller must supply a wavetable containing
trigonometric lookup tables and a workspace work.
This function converts a single real array, real_coefficient into
an equivalent complex array, complex_coefficient, (with imaginary
part set to zero), suitable for gsl_fft_complex routines. The
algorithm for the conversion is simply,
for (i = 0; i < n; i++)
{
complex_coefficient[i*stride].real
= real_coefficient[i*stride];
complex_coefficient[i*stride].imag
= 0.0;
}
This function converts halfcomplex_coefficient, an array of
half-complex coefficients as returned by gsl_fft_real_transform, into an
ordinary complex array, complex_coefficient. It fills in the
complex array using the symmetry
z_k = z_{n-k}^*
to reconstruct the redundant elements. The algorithm for the conversion
is,
complex_coefficient[0].real
= halfcomplex_coefficient[0];
complex_coefficient[0].imag
= 0.0;
for (i = 1; i < n - i; i++)
{
double hc_real
= halfcomplex_coefficient[(2 * i - 1)*stride];
double hc_imag
= halfcomplex_coefficient[(2 * i)*stride];
complex_coefficient[i*stride].real = hc_real;
complex_coefficient[i*stride].imag = hc_imag;
complex_coefficient[(n - i)*stride].real = hc_real;
complex_coefficient[(n - i)*stride].imag = -hc_imag;
}
if (i == n - i)
{
complex_coefficient[i*stride].real
= halfcomplex_coefficient[(n - 1)*stride];
complex_coefficient[i*stride].imag
= 0.0;
}
Here is an example program using gsl_fft_real_transform and
gsl_fft_halfcomplex_inverse. It generates a real signal in the
shape of a square pulse. The pulse is Fourier transformed to frequency
space, and all but the lowest ten frequency components are removed from
the array of Fourier coefficients returned by
gsl_fft_real_transform.
The remaining Fourier coefficients are transformed back to the time-domain, to give a filtered version of the square pulse. Since Fourier coefficients are stored using the half-complex symmetry both positive and negative frequencies are removed and the final filtered signal is also real.
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_real.h>
#include <gsl/gsl_fft_halfcomplex.h>
int
main (void)
{
int i, n = 100;
double data[n];
gsl_fft_real_wavetable * real;
gsl_fft_halfcomplex_wavetable * hc;
gsl_fft_real_workspace * work;
for (i = 0; i < n; i++)
{
data[i] = 0.0;
}
for (i = n / 3; i < 2 * n / 3; i++)
{
data[i] = 1.0;
}
for (i = 0; i < n; i++)
{
printf ("%d: %e\n", i, data[i]);
}
printf ("\n");
work = gsl_fft_real_workspace_alloc (n);
real = gsl_fft_real_wavetable_alloc (n);
gsl_fft_real_transform (data, 1, n,
real, work);
gsl_fft_real_wavetable_free (real);
for (i = 11; i < n; i++)
{
data[i] = 0;
}
hc = gsl_fft_halfcomplex_wavetable_alloc (n);
gsl_fft_halfcomplex_inverse (data, 1, n,
hc, work);
gsl_fft_halfcomplex_wavetable_free (hc);
for (i = 0; i < n; i++)
{
printf ("%d: %e\n", i, data[i]);
}
gsl_fft_real_workspace_free (work);
return 0;
}
Next: FFT References and Further Reading, Previous: Radix-2 FFT routines for real data, Up: Fast Fourier Transforms [Index]
Next: QAWO adaptive integration for oscillatory functions, Previous: QAWC adaptive integration for Cauchy principal values, Up: Numerical Integration [Index]
The QAWS algorithm is designed for integrands with algebraic-logarithmic singularities at the end-points of an integration region. In order to work efficiently the algorithm requires a precomputed table of Chebyshev moments.
This function allocates space for a gsl_integration_qaws_table
struct describing a singular weight function
W(x) with the parameters (\alpha, \beta, \mu, \nu),
W(x) = (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x)
where \alpha > -1, \beta > -1, and \mu = 0, 1, \nu = 0, 1. The weight function can take four different forms depending on the values of \mu and \nu,
W(x) = (x-a)^alpha (b-x)^beta (mu = 0, nu = 0) W(x) = (x-a)^alpha (b-x)^beta log(x-a) (mu = 1, nu = 0) W(x) = (x-a)^alpha (b-x)^beta log(b-x) (mu = 0, nu = 1) W(x) = (x-a)^alpha (b-x)^beta log(x-a) log(b-x) (mu = 1, nu = 1)
The singular points (a,b) do not have to be specified until the integral is computed, where they are the endpoints of the integration range.
The function returns a pointer to the newly allocated table
gsl_integration_qaws_table if no errors were detected, and 0 in
the case of error.
This function modifies the parameters (\alpha, \beta, \mu, \nu) of
an existing gsl_integration_qaws_table struct t.
This function frees all the memory associated with the
gsl_integration_qaws_table struct t.
This function computes the integral of the function f(x) over the interval (a,b) with the singular weight function (x-a)^\alpha (b-x)^\beta \log^\mu (x-a) \log^\nu (b-x). The parameters of the weight function (\alpha, \beta, \mu, \nu) are taken from the table t. The integral is,
I = \int_a^b dx f(x) (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x).
The adaptive bisection algorithm of QAG is used. When a subinterval contains one of the endpoints then a special 25-point modified Clenshaw-Curtis rule is used to control the singularities. For subintervals which do not include the endpoints an ordinary 15-point Gauss-Kronrod integration rule is used.
Next: QAWO adaptive integration for oscillatory functions, Previous: QAWC adaptive integration for Cauchy principal values, Up: Numerical Integration [Index]
Next: Infinities and Not-a-number, Up: Mathematical Functions [Index]
The library ensures that the standard BSD mathematical constants are defined. For reference, here is a list of the constants:
M_EThe base of exponentials, e
M_LOG2EThe base-2 logarithm of e, \log_2 (e)
M_LOG10EThe base-10 logarithm of e, \log_10 (e)
M_SQRT2The square root of two, \sqrt 2
M_SQRT1_2The square root of one-half, \sqrt{1/2}
M_SQRT3The square root of three, \sqrt 3
M_PIThe constant pi, \pi
M_PI_2Pi divided by two, \pi/2
M_PI_4Pi divided by four, \pi/4
M_SQRTPIThe square root of pi, \sqrt\pi
M_2_SQRTPITwo divided by the square root of pi, 2/\sqrt\pi
M_1_PIThe reciprocal of pi, 1/\pi
M_2_PITwice the reciprocal of pi, 2/\pi
M_LN10The natural logarithm of ten, \ln(10)
M_LN2The natural logarithm of two, \ln(2)
M_LNPIThe natural logarithm of pi, \ln(\pi)
M_EULEREuler’s constant, \gamma
Previous: Numerical integration examples, Up: Numerical Integration [Index]
The following book is the definitive reference for QUADPACK, and was written by the original authors. It provides descriptions of the algorithms, program listings, test programs and examples. It also includes useful advice on numerical integration and many references to the numerical integration literature used in developing QUADPACK.
The CQUAD integration algorithm is described in the following paper:
Next: Numerical integration error codes, Previous: CQUAD doubly-adaptive integration, Up: Numerical Integration [Index]
The fixed-order Gauss-Legendre integration routines are provided for fast integration of smooth functions with known polynomial order. The n-point Gauss-Legendre rule is exact for polynomials of order 2*n-1 or less. For example, these rules are useful when integrating basis functions to form mass matrices for the Galerkin method. Unlike other numerical integration routines within the library, these routines do not accept absolute or relative error bounds.
This function determines the Gauss-Legendre abscissae and weights necessary for an n-point fixed order integration scheme. If possible, high precision precomputed coefficients are used. If precomputed weights are not available, lower precision coefficients are computed on the fly.
This function applies the Gauss-Legendre integration rule contained in table t and returns the result.
For i in [0, …, t->n - 1], this function obtains the i-th Gauss-Legendre point xi and weight wi on the interval [a,b]. The points and weights are ordered by increasing point value. A function f may be integrated on [a,b] by summing wi * f(xi) over i.
This function frees the memory associated with the table t.
Next: Nonlinear Least-Squares Large Example, Previous: Nonlinear Least-Squares Geodesic Acceleration Example, Up: Nonlinear Least-Squares Examples [Index]
The following program compares all available nonlinear least squares trust-region subproblem (TRS) methods on the Branin function, a common optimization test problem. The cost function is given by
\Phi(x) &= 1/2 (f_1^2 + f_2^2) f_1 &= x_2 + a_1 x_1^2 + a_2 x_1 + a_3 f_2 &= sqrt(a_4) sqrt(1 + (1 - a_5) cos(x_1))
with a_1 = -{5.1 \over 4 \pi^2}, a_2 = {5 \over \pi}, a_3 = -6, a_4 = 10, a_5 = {1 \over 8\pi}. There are three minima of this function in the range (x_1,x_2) \in [-5,15] \times [-5,15]. The program below uses the starting point (x_1,x_2) = (6,14.5) and calculates the solution with all available nonlinear least squares TRS methods. The program output is shown below.
Method NITER NFEV NJEV Initial Cost Final cost Final cond(J) Final x levenberg-marquardt 20 27 21 1.9874e+02 3.9789e-01 6.1399e+07 (-3.14e+00, 1.23e+01) levenberg-marquardt+accel 27 36 28 1.9874e+02 3.9789e-01 1.4465e+07 (3.14e+00, 2.27e+00) dogleg 23 64 23 1.9874e+02 3.9789e-01 5.0692e+08 (3.14e+00, 2.28e+00) double-dogleg 24 69 24 1.9874e+02 3.9789e-01 3.4879e+07 (3.14e+00, 2.27e+00) 2D-subspace 23 54 24 1.9874e+02 3.9789e-01 2.5142e+07 (3.14e+00, 2.27e+00)
The first row of output above corresponds to standard Levenberg-Marquardt, while the second row includes geodesic acceleration. We see that the standard LM method converges to the minimum at (-\pi,12.275) and also uses the least number of iterations and Jacobian evaluations. All other methods converge to the minimum (\pi,2.275) and perform similarly in terms of number of Jacobian evaluations. We see that J is fairly ill-conditioned at both minima, indicating that the QR (or SVD) solver is the best choice for this problem. Since there are only two parameters in this optimization problem, we can easily visualize the paths taken by each method, which are shown in the figure below. The figure shows contours of the cost function \Phi(x_1,x_2) which exhibits three global minima in the range [-5,15] \times [-5,15]. The paths taken by each solver are shown as colored lines.
The program is given below.
#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_multifit_nlinear.h>
/* parameters to model */
struct model_params
{
double a1;
double a2;
double a3;
double a4;
double a5;
};
/* Branin function */
int
func_f (const gsl_vector * x, void *params, gsl_vector * f)
{
struct model_params *par = (struct model_params *) params;
double x1 = gsl_vector_get(x, 0);
double x2 = gsl_vector_get(x, 1);
double f1 = x2 + par->a1 * x1 * x1 + par->a2 * x1 + par->a3;
double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * cos(x1));
gsl_vector_set(f, 0, f1);
gsl_vector_set(f, 1, f2);
return GSL_SUCCESS;
}
int
func_df (const gsl_vector * x, void *params, gsl_matrix * J)
{
struct model_params *par = (struct model_params *) params;
double x1 = gsl_vector_get(x, 0);
double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * cos(x1));
gsl_matrix_set(J, 0, 0, 2.0 * par->a1 * x1 + par->a2);
gsl_matrix_set(J, 0, 1, 1.0);
gsl_matrix_set(J, 1, 0, -0.5 * par->a4 / f2 * (1.0 - par->a5) * sin(x1));
gsl_matrix_set(J, 1, 1, 0.0);
return GSL_SUCCESS;
}
int
func_fvv (const gsl_vector * x, const gsl_vector * v,
void *params, gsl_vector * fvv)
{
struct model_params *par = (struct model_params *) params;
double x1 = gsl_vector_get(x, 0);
double v1 = gsl_vector_get(v, 0);
double c = cos(x1);
double s = sin(x1);
double f2 = sqrt(par->a4) * sqrt(1.0 + (1.0 - par->a5) * c);
double t = 0.5 * par->a4 * (1.0 - par->a5) / f2;
gsl_vector_set(fvv, 0, 2.0 * par->a1 * v1 * v1);
gsl_vector_set(fvv, 1, -t * (c + s*s/f2) * v1 * v1);
return GSL_SUCCESS;
}
void
callback(const size_t iter, void *params,
const gsl_multifit_nlinear_workspace *w)
{
gsl_vector * x = gsl_multifit_nlinear_position(w);
double x1 = gsl_vector_get(x, 0);
double x2 = gsl_vector_get(x, 1);
/* print out current location */
printf("%f %f\n", x1, x2);
}
void
solve_system(gsl_vector *x0, gsl_multifit_nlinear_fdf *fdf,
gsl_multifit_nlinear_parameters *params)
{
const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
const size_t max_iter = 200;
const double xtol = 1.0e-8;
const double gtol = 1.0e-8;
const double ftol = 1.0e-8;
const size_t n = fdf->n;
const size_t p = fdf->p;
gsl_multifit_nlinear_workspace *work =
gsl_multifit_nlinear_alloc(T, params, n, p);
gsl_vector * f = gsl_multifit_nlinear_residual(work);
gsl_vector * x = gsl_multifit_nlinear_position(work);
int info;
double chisq0, chisq, rcond;
printf("# %s/%s\n",
gsl_multifit_nlinear_name(work),
gsl_multifit_nlinear_trs_name(work));
/* initialize solver */
gsl_multifit_nlinear_init(x0, fdf, work);
/* store initial cost */
gsl_blas_ddot(f, f, &chisq0);
/* iterate until convergence */
gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
callback, NULL, &info, work);
/* store final cost */
gsl_blas_ddot(f, f, &chisq);
/* store cond(J(x)) */
gsl_multifit_nlinear_rcond(&rcond, work);
/* print summary */
fprintf(stderr, "%-25s %-6zu %-5zu %-5zu %-13.4e %-12.4e %-13.4e (%.2e, %.2e)\n",
gsl_multifit_nlinear_trs_name(work),
gsl_multifit_nlinear_niter(work),
fdf->nevalf,
fdf->nevaldf,
chisq0,
chisq,
1.0 / rcond,
gsl_vector_get(x, 0),
gsl_vector_get(x, 1));
printf("\n\n");
gsl_multifit_nlinear_free(work);
}
int
main (void)
{
const size_t n = 2;
const size_t p = 2;
gsl_vector *f = gsl_vector_alloc(n);
gsl_vector *x = gsl_vector_alloc(p);
gsl_multifit_nlinear_fdf fdf;
gsl_multifit_nlinear_parameters fdf_params =
gsl_multifit_nlinear_default_parameters();
struct model_params params;
params.a1 = -5.1 / (4.0 * M_PI * M_PI);
params.a2 = 5.0 / M_PI;
params.a3 = -6.0;
params.a4 = 10.0;
params.a5 = 1.0 / (8.0 * M_PI);
/* print map of Phi(x1, x2) */
{
double x1, x2, chisq;
for (x1 = -5.0; x1 < 15.0; x1 += 0.1)
{
for (x2 = -5.0; x2 < 15.0; x2 += 0.1)
{
gsl_vector_set(x, 0, x1);
gsl_vector_set(x, 1, x2);
func_f(x, ¶ms, f);
gsl_blas_ddot(f, f, &chisq);
printf("%f %f %f\n", x1, x2, chisq);
}
printf("\n");
}
printf("\n\n");
}
/* define function to be minimized */
fdf.f = func_f;
fdf.df = func_df;
fdf.fvv = func_fvv;
fdf.n = n;
fdf.p = p;
fdf.params = ¶ms;
/* starting point */
gsl_vector_set(x, 0, 6.0);
gsl_vector_set(x, 1, 14.5);
fprintf(stderr, "%-25s %-6s %-5s %-5s %-13s %-12s %-13s %-15s\n",
"Method", "NITER", "NFEV", "NJEV", "Initial Cost",
"Final cost", "Final cond(J)", "Final x");
fdf_params.trs = gsl_multifit_nlinear_trs_lm;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multifit_nlinear_trs_dogleg;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multifit_nlinear_trs_ddogleg;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multifit_nlinear_trs_subspace2D;
solve_system(x, &fdf, &fdf_params);
gsl_vector_free(f);
gsl_vector_free(x);
return 0;
}
Next: Nonlinear Least-Squares Large Example, Previous: Nonlinear Least-Squares Geodesic Acceleration Example, Up: Nonlinear Least-Squares Examples [Index]
Previous: Example programs for Multidimensional Root finding, Up: Multidimensional Root-Finding [Index]
The original version of the Hybrid method is described in the following articles by Powell,
The following papers are also relevant to the algorithms described in this section,
Next: QAGS adaptive integration with singularities, Previous: QNG non-adaptive Gauss-Kronrod integration, Up: Numerical Integration [Index]
The QAG algorithm is a simple adaptive integration procedure. The
integration region is divided into subintervals, and on each iteration
the subinterval with the largest estimated error is bisected. This
reduces the overall error rapidly, as the subintervals become
concentrated around local difficulties in the integrand. These
subintervals are managed by a gsl_integration_workspace struct,
which handles the memory for the subinterval ranges, results and error
estimates.
This function allocates a workspace sufficient to hold n double precision intervals, their integration results and error estimates. One workspace may be used multiple times as all necessary reinitialization is performed automatically by the integration routines.
This function frees the memory associated with the workspace w.
This function applies an integration rule adaptively until an estimate of the integral of f over (a,b) is achieved within the desired absolute and relative error limits, epsabs and epsrel. The function returns the final approximation, result, and an estimate of the absolute error, abserr. The integration rule is determined by the value of key, which should be chosen from the following symbolic names,
GSL_INTEG_GAUSS15 (key = 1) GSL_INTEG_GAUSS21 (key = 2) GSL_INTEG_GAUSS31 (key = 3) GSL_INTEG_GAUSS41 (key = 4) GSL_INTEG_GAUSS51 (key = 5) GSL_INTEG_GAUSS61 (key = 6)
corresponding to the 15, 21, 31, 41, 51 and 61 point Gauss-Kronrod rules. The higher-order rules give better accuracy for smooth functions, while lower-order rules save time when the function contains local difficulties, such as discontinuities.
On each iteration the adaptive integration strategy bisects the interval with the largest error estimate. The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals is given by limit, which may not exceed the allocated size of the workspace.
Next: QAGS adaptive integration with singularities, Previous: QNG non-adaptive Gauss-Kronrod integration, Up: Numerical Integration [Index]
Next: Algorithms without Derivatives, Previous: Search Stopping Parameters for the multidimensional solver, Up: Multidimensional Root-Finding [Index]
The root finding algorithms described in this section make use of both the function and its derivative. They require an initial guess for the location of the root, but there is no absolute guarantee of convergence—the function must be suitable for this technique and the initial guess must be sufficiently close to the root for it to work. When the conditions are satisfied then convergence is quadratic.
This is a modified version of Powell’s Hybrid method as implemented in the HYBRJ algorithm in MINPACK. Minpack was written by Jorge J. Moré, Burton S. Garbow and Kenneth E. Hillstrom. The Hybrid algorithm retains the fast convergence of Newton’s method but will also reduce the residual when Newton’s method is unreliable.
The algorithm uses a generalized trust region to keep each step under control. In order to be accepted a proposed new position x' must satisfy the condition |D (x' - x)| < \delta, where D is a diagonal scaling matrix and \delta is the size of the trust region. The components of D are computed internally, using the column norms of the Jacobian to estimate the sensitivity of the residual to each component of x. This improves the behavior of the algorithm for badly scaled functions.
On each iteration the algorithm first determines the standard Newton step by solving the system J dx = - f. If this step falls inside the trust region it is used as a trial step in the next stage. If not, the algorithm uses the linear combination of the Newton and gradient directions which is predicted to minimize the norm of the function while staying inside the trust region,
dx = - \alpha J^{-1} f(x) - \beta \nabla |f(x)|^2.
This combination of Newton and gradient directions is referred to as a dogleg step.
The proposed step is now tested by evaluating the function at the resulting point, x'. If the step reduces the norm of the function sufficiently then it is accepted and size of the trust region is increased. If the proposed step fails to improve the solution then the size of the trust region is decreased and another trial step is computed.
The speed of the algorithm is increased by computing the changes to the Jacobian approximately, using a rank-1 update. If two successive attempts fail to reduce the residual then the full Jacobian is recomputed. The algorithm also monitors the progress of the solution and returns an error if several steps fail to make any improvement,
GSL_ENOPROGthe iteration is not making any progress, preventing the algorithm from continuing.
GSL_ENOPROGJre-evaluations of the Jacobian indicate that the iteration is not making any progress, preventing the algorithm from continuing.
This algorithm is an unscaled version of hybridsj. The steps are
controlled by a spherical trust region |x' - x| < \delta, instead
of a generalized region. This can be useful if the generalized region
estimated by hybridsj is inappropriate.
Newton’s Method is the standard root-polishing algorithm. The algorithm begins with an initial guess for the location of the solution. On each iteration a linear approximation to the function F is used to estimate the step which will zero all the components of the residual. The iteration is defined by the following sequence,
x -> x' = x - J^{-1} f(x)
where the Jacobian matrix J is computed from the derivative functions provided by f. The step dx is obtained by solving the linear system,
J dx = - f(x)
using LU decomposition. If the Jacobian matrix is singular, an error
code of GSL_EDOM is returned.
This is a modified version of Newton’s method which attempts to improve global convergence by requiring every step to reduce the Euclidean norm of the residual, |f(x)|. If the Newton step leads to an increase in the norm then a reduced step of relative size,
t = (\sqrt(1 + 6 r) - 1) / (3 r)
is proposed, with r being the ratio of norms |f(x')|^2/|f(x)|^2. This procedure is repeated until a suitable step size is found.
Next: Algorithms without Derivatives, Previous: Search Stopping Parameters for the multidimensional solver, Up: Multidimensional Root-Finding [Index]
Next: Prefixes, Previous: Radioactivity, Up: Physical Constants [Index]
GSL_CONST_MKSA_NEWTONThe SI unit of force, 1 Newton.
GSL_CONST_MKSA_DYNEThe force of 1 Dyne = 10^-5 Newton.
GSL_CONST_MKSA_JOULEThe SI unit of energy, 1 Joule.
GSL_CONST_MKSA_ERGThe energy 1 erg = 10^-7 Joule.
Next: Multiset allocation, Up: Multisets [Index]
A multiset is defined by a structure containing three components, the
values of n and k, and a pointer to the multiset array.
The elements of the multiset array are all of type size_t, and
are stored in increasing order. The gsl_multiset structure
looks like this,
typedef struct
{
size_t n;
size_t k;
size_t *data;
} gsl_multiset;
Next: Auxiliary Functions for Chebyshev Series, Previous: Chebyshev Definitions, Up: Chebyshev Approximations [Index]
This function allocates space for a Chebyshev series of order n
and returns a pointer to a new gsl_cheb_series struct.
This function frees a previously allocated Chebyshev series cs.
This function computes the Chebyshev approximation cs for the function f over the range (a,b) to the previously specified order. The computation of the Chebyshev approximation is an O(n^2) process, and requires n function evaluations.
Next: Random number generator initialization, Previous: General comments on random numbers, Up: Random Number Generation [Index]
It is important to remember that a random number generator is not a “real” function like sine or cosine. Unlike real functions, successive calls to a random number generator yield different return values. Of course that is just what you want for a random number generator, but to achieve this effect, the generator must keep track of some kind of “state” variable. Sometimes this state is just an integer (sometimes just the value of the previously generated random number), but often it is more complicated than that and may involve a whole array of numbers, possibly with some indices thrown in. To use the random number generators, you do not need to know the details of what comprises the state, and besides that varies from algorithm to algorithm.
The random number generator library uses two special structs,
gsl_rng_type which holds static information about each type of
generator and gsl_rng which describes an instance of a generator
created from a given gsl_rng_type.
The functions described in this section are declared in the header file gsl_rng.h.
gsl-ref-html-2.3/Multisets.html 0000664 0001750 0001750 00000013173 13055414417 014623 0 ustar edd eddNext: Sorting, Previous: Combinations, Up: Top [Index]
This chapter describes functions for creating and manipulating multisets. A multiset c is represented by an array of k integers in the range 0 to n-1, where each value c_i may occur more than once. The multiset c corresponds to indices of k elements chosen from an n element vector with replacement. In mathematical terms, n is the cardinality of the multiset while k is the maximum multiplicity of any value. Multisets are useful, for example, when iterating over the indices of a k-th order symmetric tensor in n-space.
The functions described in this chapter are defined in the header file gsl_multiset.h.
| • The Multiset struct: | ||
| • Multiset allocation: | ||
| • Accessing multiset elements: | ||
| • Multiset properties: | ||
| • Multiset functions: | ||
| • Reading and writing multisets: | ||
| • Multiset Examples: |
Next: Quasi-random number generator examples, Previous: Saving and restoring quasi-random number generator state, Up: Quasi-Random Sequences [Index]
The following quasi-random sequence algorithms are available,
This generator uses the algorithm described in Bratley, Fox, Niederreiter, ACM Trans. Model. Comp. Sim. 2, 195 (1992). It is valid up to 12 dimensions.
This generator uses the Sobol sequence described in Antonov, Saleev, USSR Comput. Maths. Math. Phys. 19, 252 (1980). It is valid up to 40 dimensions.
These generators use the Halton and reverse Halton sequences described in J.H. Halton, Numerische Mathematik 2, 84-90 (1960) and B. Vandewoestyne and R. Cools Computational and Applied Mathematics 189, 1&2, 341-361 (2006). They are valid up to 1229 dimensions.
Next: Radix-2 FFT routines for real data, Previous: Mixed-radix FFT routines for complex data, Up: Fast Fourier Transforms [Index]
The functions for real data are similar to those for complex data. However, there is an important difference between forward and inverse transforms. The Fourier transform of a real sequence is not real. It is a complex sequence with a special symmetry:
z_k = z_{n-k}^*
A sequence with this symmetry is called conjugate-complex or
half-complex. This different structure requires different
storage layouts for the forward transform (from real to half-complex)
and inverse transform (from half-complex back to real). As a
consequence the routines are divided into two sets: functions in
gsl_fft_real which operate on real sequences and functions in
gsl_fft_halfcomplex which operate on half-complex sequences.
Functions in gsl_fft_real compute the frequency coefficients of a
real sequence. The half-complex coefficients c of a real sequence
x are given by Fourier analysis,
c_k = \sum_{j=0}^{n-1} x_j \exp(-2 \pi i j k /n)
Functions in gsl_fft_halfcomplex compute inverse or backwards
transforms. They reconstruct real sequences by Fourier synthesis from
their half-complex frequency coefficients, c,
x_j = {1 \over n} \sum_{k=0}^{n-1} c_k \exp(2 \pi i j k /n)
The symmetry of the half-complex sequence implies that only half of the complex numbers in the output need to be stored. The remaining half can be reconstructed using the half-complex symmetry condition. This works for all lengths, even and odd—when the length is even the middle value where k=n/2 is also real. Thus only n real numbers are required to store the half-complex sequence, and the transform of a real sequence can be stored in the same size array as the original data.
The precise storage arrangements depend on the algorithm, and are different for radix-2 and mixed-radix routines. The radix-2 function operates in-place, which constrains the locations where each element can be stored. The restriction forces real and imaginary parts to be stored far apart. The mixed-radix algorithm does not have this restriction, and it stores the real and imaginary parts of a given term in neighboring locations (which is desirable for better locality of memory accesses).
Next: Radix-2 FFT routines for real data, Previous: Mixed-radix FFT routines for complex data, Up: Fast Fourier Transforms [Index]
Next: Computing the rank, Previous: Sorting vectors, Up: Sorting [Index]
The functions described in this section select the k smallest or largest elements of a data set of size N. The routines use an O(kN) direct insertion algorithm which is suited to subsets that are small compared with the total size of the dataset. For example, the routines are useful for selecting the 10 largest values from one million data points, but not for selecting the largest 100,000 values. If the subset is a significant part of the total dataset it may be faster to sort all the elements of the dataset directly with an O(N \log N) algorithm and obtain the smallest or largest values that way.
This function copies the k smallest elements of the array src, of size n and stride stride, in ascending numerical order into the array dest. The size k of the subset must be less than or equal to n. The data src is not modified by this operation.
This function copies the k largest elements of the array src, of size n and stride stride, in descending numerical order into the array dest. k must be less than or equal to n. The data src is not modified by this operation.
These functions copy the k smallest or largest elements of the vector v into the array dest. k must be less than or equal to the length of the vector v.
The following functions find the indices of the k smallest or largest elements of a dataset,
This function stores the indices of the k smallest elements of the array src, of size n and stride stride, in the array p. The indices are chosen so that the corresponding data is in ascending numerical order. k must be less than or equal to n. The data src is not modified by this operation.
This function stores the indices of the k largest elements of the array src, of size n and stride stride, in the array p. The indices are chosen so that the corresponding data is in descending numerical order. k must be less than or equal to n. The data src is not modified by this operation.
These functions store the indices of the k smallest or largest elements of the vector v in the array p. k must be less than or equal to the length of the vector v.
Next: Computing the rank, Previous: Sorting vectors, Up: Sorting [Index]
Next: Regular Bessel Function - Fractional Order, Previous: Regular Modified Spherical Bessel Functions, Up: Bessel Functions [Index]
The irregular modified spherical Bessel functions k_l(x) are related to the irregular modified Bessel functions of fractional order, k_l(x) = \sqrt{\pi/(2x)} K_{l+1/2}(x).
These routines compute the scaled irregular modified spherical Bessel function of zeroth order, \exp(x) k_0(x), for x>0.
These routines compute the scaled irregular modified spherical Bessel function of first order, \exp(x) k_1(x), for x>0.
These routines compute the scaled irregular modified spherical Bessel function of second order, \exp(x) k_2(x), for x>0.
These routines compute the scaled irregular modified spherical Bessel function of order l, \exp(x) k_l(x), for x>0.
This routine computes the values of the scaled irregular modified spherical Bessel functions \exp(x) k_l(x) for l from 0 to lmax inclusive for lmax >= 0 and x>0, storing the results in the array result_array. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
Next: Variable Index, Previous: GNU Free Documentation License, Up: Top [Index]
| Jump to: | C G |
|---|
| Jump to: | C G |
|---|
Next: Variable Index, Previous: GNU Free Documentation License, Up: Top [Index]
Next: Example programs for vectors, Previous: Finding maximum and minimum elements of vectors, Up: Vectors [Index]
The following functions are defined for real and complex vectors. For complex vectors both the real and imaginary parts must satisfy the conditions.
These functions return 1 if all the elements of the vector v are zero, strictly positive, strictly negative, or non-negative respectively, and 0 otherwise.
This function returns 1 if the vectors u and v are equal (by comparison of element values) and 0 otherwise.
Next: Maximum and Minimum functions, Previous: Testing the Sign of Numbers, Up: Mathematical Functions [Index]
This macro evaluates to 1 if n is odd and 0 if n is even. The argument n must be of integer type.
This macro is the opposite of GSL_IS_ODD(n). It evaluates to 1 if
n is even and 0 if n is odd. The argument n must be of
integer type.
Previous: Legendre Form of Incomplete Elliptic Integrals, Up: Elliptic Integrals [Index]
These routines compute the incomplete elliptic integral RC(x,y) to the accuracy specified by the mode variable mode.
These routines compute the incomplete elliptic integral RD(x,y,z) to the accuracy specified by the mode variable mode.
These routines compute the incomplete elliptic integral RF(x,y,z) to the accuracy specified by the mode variable mode.
These routines compute the incomplete elliptic integral RJ(x,y,z,p) to the accuracy specified by the mode variable mode.
Next: Triangular Systems, Previous: Householder solver for linear systems, Up: Linear Algebra [Index]
The functions described in this section efficiently solve symmetric,
non-symmetric and cyclic tridiagonal systems with minimal storage.
Note that the current implementations of these functions use a variant
of Cholesky decomposition, so the tridiagonal matrix must be positive
definite. For non-positive definite matrices, the functions return
the error code GSL_ESING.
This function solves the general N-by-N system A x = b where A is tridiagonal (N >= 2). The super-diagonal and sub-diagonal vectors e and f must be one element shorter than the diagonal vector diag. The form of A for the 4-by-4 case is shown below,
A = ( d_0 e_0 0 0 )
( f_0 d_1 e_1 0 )
( 0 f_1 d_2 e_2 )
( 0 0 f_2 d_3 )
This function solves the general N-by-N system A x = b where A is symmetric tridiagonal (N >= 2). The off-diagonal vector e must be one element shorter than the diagonal vector diag. The form of A for the 4-by-4 case is shown below,
A = ( d_0 e_0 0 0 )
( e_0 d_1 e_1 0 )
( 0 e_1 d_2 e_2 )
( 0 0 e_2 d_3 )
This function solves the general N-by-N system A x = b where A is cyclic tridiagonal (N >= 3). The cyclic super-diagonal and sub-diagonal vectors e and f must have the same number of elements as the diagonal vector diag. The form of A for the 4-by-4 case is shown below,
A = ( d_0 e_0 0 f_3 )
( f_0 d_1 e_1 0 )
( 0 f_1 d_2 e_2 )
( e_3 0 f_2 d_3 )
This function solves the general N-by-N system A x = b where A is symmetric cyclic tridiagonal (N >= 3). The cyclic off-diagonal vector e must have the same number of elements as the diagonal vector diag. The form of A for the 4-by-4 case is shown below,
A = ( d_0 e_0 0 e_3 )
( e_0 d_1 e_1 0 )
( 0 e_1 d_2 e_2 )
( e_3 0 e_2 d_3 )
Next: Triangular Systems, Previous: Householder solver for linear systems, Up: Linear Algebra [Index]
Next: Median and Percentiles, Previous: Weighted Samples, Up: Statistics [Index]
The following functions find the maximum and minimum values of a
dataset (or their indices). If the data contains NaNs then a
NaN will be returned, since the maximum or minimum value is
undefined. For functions which return an index, the location of the
first NaN in the array is returned.
This function returns the maximum value in data, a dataset of length n with stride stride. The maximum value is defined as the value of the element x_i which satisfies x_i >= x_j for all j.
If you want instead to find the element with the largest absolute
magnitude you will need to apply fabs or abs to your data
before calling this function.
This function returns the minimum value in data, a dataset of length n with stride stride. The minimum value is defined as the value of the element x_i which satisfies x_i <= x_j for all j.
If you want instead to find the element with the smallest absolute
magnitude you will need to apply fabs or abs to your data
before calling this function.
This function finds both the minimum and maximum values min, max in data in a single pass.
This function returns the index of the maximum value in data, a dataset of length n with stride stride. The maximum value is defined as the value of the element x_i which satisfies x_i >= x_j for all j. When there are several equal maximum elements then the first one is chosen.
This function returns the index of the minimum value in data, a dataset of length n with stride stride. The minimum value is defined as the value of the element x_i which satisfies x_i >= x_j for all j. When there are several equal minimum elements then the first one is chosen.
This function returns the indexes min_index, max_index of the minimum and maximum values in data in a single pass.
Next: Median and Percentiles, Previous: Weighted Samples, Up: Statistics [Index]
Next: Mathematical Functions, Previous: Using the library, Up: Top [Index]
This chapter describes the way that GSL functions report and handle errors. By examining the status information returned by every function you can determine whether it succeeded or failed, and if it failed you can find out what the precise cause of failure was. You can also define your own error handling functions to modify the default behavior of the library.
The functions described in this section are declared in the header file gsl_errno.h.
| • Error Reporting: | ||
| • Error Codes: | ||
| • Error Handlers: | ||
| • Using GSL error reporting in your own functions: | ||
| • Error Reporting Examples: |
Next: Angular Mathieu Functions, Previous: Mathieu Function Workspace, Up: Mathieu Functions [Index]
These routines compute the characteristic values a_n(q), b_n(q) of the Mathieu functions ce_n(q,x) and se_n(q,x), respectively.
These routines compute a series of Mathieu characteristic values a_n(q), b_n(q) for n from order_min to order_max inclusive, storing the results in the array result_array.
Next: Thread-safety, Previous: Compatibility with C++, Up: Using the library [Index]
The library assumes that arrays, vectors and matrices passed as
modifiable arguments are not aliased and do not overlap with each other.
This removes the need for the library to handle overlapping memory
regions as a special case, and allows additional optimizations to be
used. If overlapping memory regions are passed as modifiable arguments
then the results of such functions will be undefined. If the arguments
will not be modified (for example, if a function prototype declares them
as const arguments) then overlapping or aliased memory regions
can be safely used.
Next: Real Generalized Nonsymmetric Eigensystems, Previous: Real Generalized Symmetric-Definite Eigensystems, Up: Eigensystems [Index]
The complex generalized hermitian-definite eigenvalue problem is to find eigenvalues \lambda and eigenvectors x such that
A x = \lambda B x
where A and B are hermitian matrices, and B is positive-definite. Similarly to the real case, this can be reduced to C y = \lambda y where C = L^{-1} A L^{-H} is hermitian, and y = L^H x. The standard hermitian eigensolver can be applied to the matrix C. The resulting eigenvectors are backtransformed to find the vectors of the original problem. The eigenvalues of the generalized hermitian-definite eigenproblem are always real.
This function allocates a workspace for computing eigenvalues of n-by-n complex generalized hermitian-definite eigensystems. The size of the workspace is O(3n).
This function frees the memory associated with the workspace w.
This function computes the eigenvalues of the complex generalized hermitian-definite matrix pair (A, B), and stores them in eval, using the method outlined above. On output, B contains its Cholesky decomposition and A is destroyed.
This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n complex generalized hermitian-definite eigensystems. The size of the workspace is O(5n).
This function frees the memory associated with the workspace w.
This function computes the eigenvalues and eigenvectors of the complex generalized hermitian-definite matrix pair (A, B), and stores them in eval and evec respectively. The computed eigenvectors are normalized to have unit magnitude. On output, B contains its Cholesky decomposition and A is destroyed.
Next: Real Generalized Nonsymmetric Eigensystems, Previous: Real Generalized Symmetric-Definite Eigensystems, Up: Eigensystems [Index]
Next: Complex Trigonometric Functions, Previous: Complex arithmetic operators, Up: Complex Numbers [Index]
This function returns the square root of the complex number z, \sqrt z. The branch cut is the negative real axis. The result always lies in the right half of the complex plane.
This function returns the complex square root of the real number x, where x may be negative.
The function returns the complex number z raised to the complex power a, z^a. This is computed as \exp(\log(z)*a) using complex logarithms and complex exponentials.
This function returns the complex number z raised to the real power x, z^x.
This function returns the complex exponential of the complex number z, \exp(z).
This function returns the complex natural logarithm (base e) of the complex number z, \log(z). The branch cut is the negative real axis.
This function returns the complex base-10 logarithm of the complex number z, \log_10 (z).
This function returns the complex base-b logarithm of the complex number z, \log_b(z). This quantity is computed as the ratio \log(z)/\log(b).
Previous: Log Complementary Error Function, Up: Error Functions [Index]
The probability functions for the Normal or Gaussian distribution are described in Abramowitz & Stegun, Section 26.2.
These routines compute the Gaussian probability density function Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2).
These routines compute the upper tail of the Gaussian probability function Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2).
The hazard function for the normal distribution, also known as the inverse Mills’ ratio, is defined as,
h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \erfc(x/\sqrt 2)
It decreases rapidly as x approaches -\infty and asymptotes to h(x) \sim x as x approaches +\infty.
These routines compute the hazard function for the normal distribution.
Next: Overview of complex data FFTs, Up: Fast Fourier Transforms [Index]
Fast Fourier Transforms are efficient algorithms for calculating the discrete Fourier transform (DFT),
x_j = \sum_{k=0}^{n-1} z_k \exp(-2\pi i j k / n)
The DFT usually arises as an approximation to the continuous Fourier transform when functions are sampled at discrete intervals in space or time. The naive evaluation of the discrete Fourier transform is a matrix-vector multiplication W\vec{z}. A general matrix-vector multiplication takes O(n^2) operations for n data-points. Fast Fourier transform algorithms use a divide-and-conquer strategy to factorize the matrix W into smaller sub-matrices, corresponding to the integer factors of the length n. If n can be factorized into a product of integers f_1 f_2 ... f_m then the DFT can be computed in O(n \sum f_i) operations. For a radix-2 FFT this gives an operation count of O(n \log_2 n).
All the FFT functions offer three types of transform: forwards, inverse and backwards, based on the same mathematical definitions. The definition of the forward Fourier transform, x = FFT(z), is,
x_j = \sum_{k=0}^{n-1} z_k \exp(-2\pi i j k / n)
and the definition of the inverse Fourier transform, x = IFFT(z), is,
z_j = {1 \over n} \sum_{k=0}^{n-1} x_k \exp(2\pi i j k / n).
The factor of 1/n makes this a true inverse. For example, a call
to gsl_fft_complex_forward followed by a call to
gsl_fft_complex_inverse should return the original data (within
numerical errors).
In general there are two possible choices for the sign of the exponential in the transform/ inverse-transform pair. GSL follows the same convention as FFTPACK, using a negative exponential for the forward transform. The advantage of this convention is that the inverse transform recreates the original function with simple Fourier synthesis. Numerical Recipes uses the opposite convention, a positive exponential in the forward transform.
The backwards FFT is simply our terminology for an unscaled version of the inverse FFT,
z^{backwards}_j = \sum_{k=0}^{n-1} x_k \exp(2\pi i j k / n).
When the overall scale of the result is unimportant it is often convenient to use the backwards FFT instead of the inverse to save unnecessary divisions.
Next: Overview of complex data FFTs, Up: Fast Fourier Transforms [Index]
Next: The Exponential Distribution, Previous: The Bivariate Gaussian Distribution, Up: Random Number Distributions [Index]
This function generates a random vector satisfying the k-dimensional multivariate Gaussian
distribution with mean \mu and variance-covariance matrix
\Sigma. On input, the k-vector \mu is given in mu, and
the Cholesky factor of the k-by-k matrix \Sigma = L L^T is
given in the lower triangle of L, as output from gsl_linalg_cholesky_decomp.
The random vector is stored in result on output. The probability distribution
for multivariate Gaussian random variates is
p(x_1,...,x_k) dx_1 ... dx_k = {1 \over \sqrt{(2 \pi)^k |\Sigma|} \exp \left(-{1 \over 2} (x - \mu)^T \Sigma^{-1} (x - \mu)\right) dx_1 \dots dx_k
These functions compute p(x) or \log{p(x)} at the point x, using mean vector mu and variance-covariance matrix specified by its Cholesky factor L using the formula above. Additional workspace of length k is required in work.
Given a set of n samples X_j from a k-dimensional multivariate Gaussian distribution, this function computes the maximum likelihood estimate of the mean of the distribution, given by
\Hat{\mu} = {1 \over n} \sum_{j=1}^n X_j
The samples X_1,X_2,\dots,X_n are given in the n-by-k matrix X, and the maximum likelihood estimate of the mean is stored in mu_hat on output.
Given a set of n samples X_j from a k-dimensional multivariate Gaussian distribution, this function computes the maximum likelihood estimate of the variance-covariance matrix of the distribution, given by
\Hat{\Sigma} = {1 \over n} \sum_{j=1}^n \left( X_j - \Hat{\mu} \right) \left( X_j - \Hat{\mu} \right)^T
The samples X_1,X_2,\dots,X_n are given in the n-by-k matrix X and the maximum likelihood estimate of the variance-covariance matrix is stored in sigma_hat on output.
Next: The Exponential Distribution, Previous: The Bivariate Gaussian Distribution, Up: Random Number Distributions [Index]
Next: Histogramming ntuple values, Previous: Reading ntuples, Up: N-tuples [Index]
This function closes the ntuple file ntuple and frees its associated allocated memory.
Previous: Using GSL error reporting in your own functions, Up: Error Handling [Index]
Here is an example of some code which checks the return value of a function where an error might be reported,
#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>
...
int status;
size_t n = 37;
gsl_set_error_handler_off();
status = gsl_fft_complex_radix2_forward (data, stride, n);
if (status) {
if (status == GSL_EINVAL) {
fprintf (stderr, "invalid argument, n=%d\n", n);
} else {
fprintf (stderr, "failed, gsl_errno=%d\n",
status);
}
exit (-1);
}
...
The function gsl_fft_complex_radix2 only accepts integer lengths
which are a power of two. If the variable n is not a power of
two then the call to the library function will return GSL_EINVAL,
indicating that the length argument is invalid. The function call to
gsl_set_error_handler_off stops the default error handler from
aborting the program. The else clause catches any other possible
errors.
Next: Elementary Operations, Previous: Debye Functions, Up: Special Functions [Index]
The functions described in this section are declared in the header file gsl_sf_dilog.h.
| • Real Argument: | ||
| • Complex Argument: |
Next: Trigamma Function, Up: Psi (Digamma) Function [Index]
These routines compute the digamma function \psi(n) for positive integer n. The digamma function is also called the Psi function.
These routines compute the digamma function \psi(x) for general x, x \ne 0.
These routines compute the real part of the digamma function on the line 1+i y, \Re[\psi(1 + i y)].
Next: Example ntuple programs, Previous: Closing an ntuple file, Up: N-tuples [Index]
Once an ntuple has been created its contents can be histogrammed in
various ways using the function gsl_ntuple_project. Two
user-defined functions must be provided, a function to select events and
a function to compute scalar values. The selection function and the
value function both accept the ntuple row as a first argument and other
parameters as a second argument.
The selection function determines which ntuple rows are selected for histogramming. It is defined by the following struct,
typedef struct {
int (* function) (void * ntuple_data, void * params);
void * params;
} gsl_ntuple_select_fn;
The struct component function should return a non-zero value for each ntuple row that is to be included in the histogram.
The value function computes scalar values for those ntuple rows selected by the selection function,
typedef struct {
double (* function) (void * ntuple_data, void * params);
void * params;
} gsl_ntuple_value_fn;
In this case the struct component function should return the value to be added to the histogram for the ntuple row.
This function updates the histogram h from the ntuple ntuple using the functions value_func and select_func. For each ntuple row where the selection function select_func is non-zero the corresponding value of that row is computed using the function value_func and added to the histogram. Those ntuple rows where select_func returns zero are ignored. New entries are added to the histogram, so subsequent calls can be used to accumulate further data in the same histogram.
Next: Example ntuple programs, Previous: Closing an ntuple file, Up: N-tuples [Index]
Next: GSL CBLAS Examples, Previous: Level 2 CBLAS Functions, Up: GSL CBLAS Library [Index]
Next: GSL CBLAS Examples, Previous: Level 2 CBLAS Functions, Up: GSL CBLAS Library [Index]
Next: Radix-2 FFT routines for complex data, Previous: Mathematical Definitions, Up: Fast Fourier Transforms [Index]
The inputs and outputs for the complex FFT routines are packed arrays of floating point numbers. In a packed array the real and imaginary parts of each complex number are placed in alternate neighboring elements. For example, the following definition of a packed array of length 6,
double x[3*2]; gsl_complex_packed_array data = x;
can be used to hold an array of three complex numbers, z[3], in
the following way,
data[0] = Re(z[0]) data[1] = Im(z[0]) data[2] = Re(z[1]) data[3] = Im(z[1]) data[4] = Re(z[2]) data[5] = Im(z[2])
The array indices for the data have the same ordering as those in the definition of the DFT—i.e. there are no index transformations or permutations of the data.
A stride parameter allows the user to perform transforms on the
elements z[stride*i] instead of z[i]. A stride greater
than 1 can be used to take an in-place FFT of the column of a matrix. A
stride of 1 accesses the array without any additional spacing between
elements.
To perform an FFT on a vector argument, such as gsl_vector_complex
* v, use the following definitions (or their equivalents) when calling
the functions described in this chapter:
gsl_complex_packed_array data = v->data; size_t stride = v->stride; size_t n = v->size;
For physical applications it is important to remember that the index appearing in the DFT does not correspond directly to a physical frequency. If the time-step of the DFT is \Delta then the frequency-domain includes both positive and negative frequencies, ranging from -1/(2\Delta) through 0 to +1/(2\Delta). The positive frequencies are stored from the beginning of the array up to the middle, and the negative frequencies are stored backwards from the end of the array.
Here is a table which shows the layout of the array data, and the correspondence between the time-domain data z, and the frequency-domain data x.
index z x = FFT(z)
0 z(t = 0) x(f = 0)
1 z(t = 1) x(f = 1/(n Delta))
2 z(t = 2) x(f = 2/(n Delta))
. ........ ..................
n/2 z(t = n/2) x(f = +1/(2 Delta),
-1/(2 Delta))
. ........ ..................
n-3 z(t = n-3) x(f = -3/(n Delta))
n-2 z(t = n-2) x(f = -2/(n Delta))
n-1 z(t = n-1) x(f = -1/(n Delta))
When n is even the location n/2 contains the most positive and negative frequencies (+1/(2 \Delta), -1/(2 \Delta)) which are equivalent. If n is odd then general structure of the table above still applies, but n/2 does not appear.
Next: Radix-2 FFT routines for complex data, Previous: Mathematical Definitions, Up: Fast Fourier Transforms [Index]
Next: Quasi-random number generator algorithms, Previous: Auxiliary quasi-random number generator functions, Up: Quasi-Random Sequences [Index]
This function copies the quasi-random sequence generator src into the pre-existing generator dest, making dest into an exact copy of src. The two generators must be of the same type.
This function returns a pointer to a newly created generator which is an exact copy of the generator q.
Previous: Trigamma Function, Up: Psi (Digamma) Function [Index]
These routines compute the polygamma function \psi^{(n)}(x) for n >= 0, x > 0.
Next: Exponentiation With Error Estimate, Previous: Exponential Function, Up: Exponential Functions [Index]
These routines compute the quantity \exp(x)-1 using an algorithm that is accurate for small x.
These routines compute the quantity (\exp(x)-1)/x using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion (\exp(x)-1)/x = 1 + x/2 + x^2/(2*3) + x^3/(2*3*4) + \dots.
These routines compute the quantity 2(\exp(x)-1-x)/x^2 using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion 2(\exp(x)-1-x)/x^2 = 1 + x/3 + x^2/(3*4) + x^3/(3*4*5) + \dots.
These routines compute the N-relative exponential, which is the
n-th generalization of the functions gsl_sf_exprel and
gsl_sf_exprel_2. The N-relative exponential is given by,
exprel_N(x) = N!/x^N (\exp(x) - \sum_{k=0}^{N-1} x^k/k!)
= 1 + x/(N+1) + x^2/((N+1)(N+2)) + ...
= 1F1 (1,1+N,x)
Next: Numerical Integration, Previous: Eigensystems, Up: Top [Index]
This chapter describes functions for performing Fast Fourier Transforms (FFTs). The library includes radix-2 routines (for lengths which are a power of two) and mixed-radix routines (which work for any length). For efficiency there are separate versions of the routines for real data and for complex data. The mixed-radix routines are a reimplementation of the FFTPACK library of Paul Swarztrauber. Fortran code for FFTPACK is available on Netlib (FFTPACK also includes some routines for sine and cosine transforms but these are currently not available in GSL). For details and derivations of the underlying algorithms consult the document GSL FFT Algorithms (see FFT References and Further Reading)
Previous: Integrands with weight functions, Up: Numerical Integration Introduction [Index]
The presence of singularities (or other behavior) in the integrand can cause slow convergence in the Chebyshev approximation. The modified Clenshaw-Curtis rules used in QUADPACK separate out several common weight functions which cause slow convergence.
These weight functions are integrated analytically against the Chebyshev polynomials to precompute modified Chebyshev moments. Combining the moments with the Chebyshev approximation to the function gives the desired integral. The use of analytic integration for the singular part of the function allows exact cancellations and substantially improves the overall convergence behavior of the integration.
gsl-ref-html-2.3/Reading-and-writing-2D-histograms.html 0000664 0001750 0001750 00000017730 13055414450 021045 0 ustar edd eddNext: Resampling from 2D histograms, Previous: 2D Histogram Operations, Up: Histograms [Index]
The library provides functions for reading and writing two dimensional histograms to a file as binary data or formatted text.
This function writes the ranges and bins of the histogram h to the
stream stream in binary format. The return value is 0 for success
and GSL_EFAILED if there was a problem writing to the file. Since
the data is written in the native binary format it may not be portable
between different architectures.
This function reads into the histogram h from the stream
stream in binary format. The histogram h must be
preallocated with the correct size since the function uses the number of
x and y bins in h to determine how many bytes to read. The return
value is 0 for success and GSL_EFAILED if there was a problem
reading from the file. The data is assumed to have been written in the
native binary format on the same architecture.
This function writes the ranges and bins of the histogram h
line-by-line to the stream stream using the format specifiers
range_format and bin_format. These should be one of the
%g, %e or %f formats for floating point
numbers. The function returns 0 for success and GSL_EFAILED if
there was a problem writing to the file. The histogram output is
formatted in five columns, and the columns are separated by spaces,
like this,
xrange[0] xrange[1] yrange[0] yrange[1] bin(0,0) xrange[0] xrange[1] yrange[1] yrange[2] bin(0,1) xrange[0] xrange[1] yrange[2] yrange[3] bin(0,2) .... xrange[0] xrange[1] yrange[ny-1] yrange[ny] bin(0,ny-1) xrange[1] xrange[2] yrange[0] yrange[1] bin(1,0) xrange[1] xrange[2] yrange[1] yrange[2] bin(1,1) xrange[1] xrange[2] yrange[1] yrange[2] bin(1,2) .... xrange[1] xrange[2] yrange[ny-1] yrange[ny] bin(1,ny-1) .... xrange[nx-1] xrange[nx] yrange[0] yrange[1] bin(nx-1,0) xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,1) xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,2) .... xrange[nx-1] xrange[nx] yrange[ny-1] yrange[ny] bin(nx-1,ny-1)
Each line contains the lower and upper limits of the bin and the contents of the bin. Since the upper limits of the each bin are the lower limits of the neighboring bins there is duplication of these values but this allows the histogram to be manipulated with line-oriented tools.
This function reads formatted data from the stream stream into the
histogram h. The data is assumed to be in the five-column format
used by gsl_histogram2d_fprintf. The histogram h must be
preallocated with the correct lengths since the function uses the sizes
of h to determine how many numbers to read. The function returns 0
for success and GSL_EFAILED if there was a problem reading from
the file.
Next: Resampling from 2D histograms, Previous: 2D Histogram Operations, Up: Histograms [Index]
Next: Permutation Examples, Previous: Reading and writing permutations, Up: Permutations [Index]
A permutation can be represented in both linear and cyclic notations. The functions described in this section convert between the two forms. The linear notation is an index mapping, and has already been described above. The cyclic notation expresses a permutation as a series of circular rearrangements of groups of elements, or cycles.
For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is replaced by 3 and 3 is replaced by 1 in a circular fashion. Cycles of different sets of elements can be combined independently, for example (1 2 3) (4 5) combines the cycle (1 2 3) with the cycle (4 5), which is an exchange of elements 4 and 5. A cycle of length one represents an element which is unchanged by the permutation and is referred to as a singleton.
It can be shown that every permutation can be decomposed into combinations of cycles. The decomposition is not unique, but can always be rearranged into a standard canonical form by a reordering of elements. The library uses the canonical form defined in Knuth’s Art of Computer Programming (Vol 1, 3rd Ed, 1997) Section 1.3.3, p.178.
The procedure for obtaining the canonical form given by Knuth is,
For example, the linear representation (2 4 3 0 1) is represented as (1 4) (0 2 3) in canonical form. The permutation corresponds to an exchange of elements 1 and 4, and rotation of elements 0, 2 and 3.
The important property of the canonical form is that it can be reconstructed from the contents of each cycle without the brackets. In addition, by removing the brackets it can be considered as a linear representation of a different permutation. In the example given above the permutation (2 4 3 0 1) would become (1 4 0 2 3). This mapping has many applications in the theory of permutations.
This function computes the canonical form of the permutation p and stores it in the output argument q.
This function converts a permutation q in canonical form back into linear form storing it in the output argument p.
This function counts the number of inversions in the permutation p. An inversion is any pair of elements that are not in order. For example, the permutation 2031 has three inversions, corresponding to the pairs (2,0) (2,1) and (3,1). The identity permutation has no inversions.
This function counts the number of cycles in the permutation p, given in linear form.
This function counts the number of cycles in the permutation q, given in canonical form.
Next: Permutation Examples, Previous: Reading and writing permutations, Up: Permutations [Index]
Next: Random Number References and Further Reading, Previous: Random Number Generator Performance, Up: Random Number Generation [Index]
The following program demonstrates the use of a random number generator to produce uniform random numbers in the range [0.0, 1.0),
#include <stdio.h>
#include <gsl/gsl_rng.h>
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
int i, n = 10;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
for (i = 0; i < n; i++)
{
double u = gsl_rng_uniform (r);
printf ("%.5f\n", u);
}
gsl_rng_free (r);
return 0;
}
Here is the output of the program,
$ ./a.out
0.99974 0.16291 0.28262 0.94720 0.23166 0.48497 0.95748 0.74431 0.54004 0.73995
The numbers depend on the seed used by the generator. The default seed
can be changed with the GSL_RNG_SEED environment variable to
produce a different stream of numbers. The generator itself can be
changed using the environment variable GSL_RNG_TYPE. Here is the
output of the program using a seed value of 123 and the
multiple-recursive generator mrg,
$ GSL_RNG_SEED=123 GSL_RNG_TYPE=mrg ./a.out
0.33050 0.86631 0.32982 0.67620 0.53391 0.06457 0.16847 0.70229 0.04371 0.86374
Previous: Complete Fermi-Dirac Integrals, Up: Fermi-Dirac Function [Index]
The incomplete Fermi-Dirac integral F_j(x,b) is given by,
F_j(x,b) := (1/\Gamma(j+1)) \int_b^\infty dt (t^j / (\Exp(t-x) + 1))
These routines compute the incomplete Fermi-Dirac integral with an index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x).
Next: Robust linear regression, Previous: Multi-parameter regression, Up: Least-Squares Fitting [Index]
Ordinary weighted least squares models seek a solution vector c which minimizes the residual
\chi^2 = || y - Xc ||_W^2
where y is the n-by-1 observation vector,
X is the n-by-p design matrix, c is
the p-by-1 solution vector,
W = diag(w_1,...,w_n) is the data weighting matrix,
and ||r||_W^2 = r^T W r.
In cases where the least squares matrix X is ill-conditioned,
small perturbations (ie: noise) in the observation vector could lead to
widely different solution vectors c.
One way of dealing with ill-conditioned matrices is to use a “truncated SVD”
in which small singular values, below some given tolerance, are discarded
from the solution. The truncated SVD method is available using the functions
gsl_multifit_linear_tsvd and gsl_multifit_wlinear_tsvd. Another way
to help solve ill-posed problems is to include a regularization term in the least squares
minimization
\chi^2 = || y - Xc ||_W^2 + \lambda^2 || L c ||^2
for a suitably chosen regularization parameter \lambda and matrix L. This type of regularization is known as Tikhonov, or ridge, regression. In some applications, L is chosen as the identity matrix, giving preference to solution vectors c with smaller norms. Including this regularization term leads to the explicit “normal equations” solution
c = ( X^T W X + \lambda^2 L^T L )^-1 X^T W y
which reduces to the ordinary least squares solution when L = 0. In practice, it is often advantageous to transform a regularized least squares system into the form
\chi^2 = || y~ - X~ c~ ||^2 + \lambda^2 || c~ ||^2
This is known as the Tikhonov “standard form” and has the normal equations solution \tilde{c} = \left( \tilde{X}^T \tilde{X} + \lambda^2 I \right)^{-1} \tilde{X}^T \tilde{y}. For an m-by-p matrix L which is full rank and has m >= p (ie: L is square or has more rows than columns), we can calculate the “thin” QR decomposition of L, and note that ||L c|| = ||R c|| since the Q factor will not change the norm. Since R is p-by-p, we can then use the transformation
X~ = sqrt(W) X R^-1 y~ = sqrt(W) y c~ = R c
to achieve the standard form. For a rectangular matrix L with m < p,
a more sophisticated approach is needed (see Hansen 1998, chapter 2.3).
In practice, the normal equations solution above is not desirable due to
numerical instabilities, and so the system is solved using the
singular value decomposition of the matrix \tilde{X}.
The matrix L is often chosen as the identity matrix, or as a first
or second finite difference operator, to ensure a smoothly varying
coefficient vector c, or as a diagonal matrix to selectively damp
each model parameter differently. If L \ne I, the user must first
convert the least squares problem to standard form using
gsl_multifit_linear_stdform1 or gsl_multifit_linear_stdform2,
solve the system, and then backtransform the solution vector to recover
the solution of the original problem (see
gsl_multifit_linear_genform1 and gsl_multifit_linear_genform2).
In many regularization problems, care must be taken when choosing the regularization parameter \lambda. Since both the residual norm ||y - X c|| and solution norm ||L c|| are being minimized, the parameter \lambda represents a tradeoff between minimizing either the residuals or the solution vector. A common tool for visualizing the comprimise between the minimization of these two quantities is known as the L-curve. The L-curve is a log-log plot of the residual norm ||y - X c|| on the horizontal axis and the solution norm ||L c|| on the vertical axis. This curve nearly always as an L shaped appearance, with a distinct corner separating the horizontal and vertical sections of the curve. The regularization parameter corresponding to this corner is often chosen as the optimal value. GSL provides routines to calculate the L-curve for all relevant regularization parameters as well as locating the corner.
Another method of choosing the regularization parameter is known as Generalized Cross Validation (GCV). This method is based on the idea that if an arbitrary element y_i is left out of the right hand side, the resulting regularized solution should predict this element accurately. This leads to choosing the parameter \lambda which minimizes the GCV function
G(\lambda) = (||y - X c_{\lambda}||^2) / Tr(I_n - X X^I)^2
where X_{\lambda}^I is the matrix which relates the solution c_{\lambda} to the right hand side y, ie: c_{\lambda} = X_{\lambda}^I y. GSL provides routines to compute the GCV curve and its minimum.
For most applications, the steps required to solve a regularized least squares problem are as follows:
These functions define a regularization matrix
L = diag(l_0,l_1,...,l_{p-1}).
The diagonal matrix element l_i is provided by the
ith element of the input vector L.
The n-by-p least squares matrix X and
vector y of length n are then
converted to standard form as described above and the parameters
(\tilde{X},\tilde{y}) are stored in Xs and ys
on output. Xs and ys have the same dimensions as
X and y. Optional data weights may be supplied in the
vector w of length n. In order to apply this transformation,
L^{-1} must exist and so none of the l_i
may be zero. After the standard form system has been solved,
use gsl_multifit_linear_genform1 to recover the original solution vector.
It is allowed to have X = Xs and y = ys for an in-place transform.
In order to perform a weighted regularized fit with L = I, the user may
call gsl_multifit_linear_applyW to convert to standard form.
This function factors the m-by-p regularization matrix
L into a form needed for the later transformation to standard form. L
may have any number of rows m. If m \ge p the QR decomposition of
L is computed and stored in L on output. If m < p, the QR decomposition
of L^T is computed and stored in L on output. On output,
the Householder scalars are stored in the vector tau of size MIN(m,p).
These outputs will be used by gsl_multifit_linear_wstdform2 to complete the
transformation to standard form.
These functions convert the least squares system (X,y,W,L) to standard
form (\tilde{X},\tilde{y}) which are stored in Xs and ys
respectively. The m-by-p regularization matrix L is specified by the inputs
LQR and Ltau, which are outputs from gsl_multifit_linear_L_decomp.
The dimensions of the standard form parameters (\tilde{X},\tilde{y})
depend on whether m is larger or less than p. For m \ge p,
Xs is n-by-p, ys is n-by-1, and M is
not used. For m < p, Xs is (n - p + m)-by-m,
ys is (n - p + m)-by-1, and M is additional n-by-p workspace,
which is required to recover the original solution vector after the system has been
solved (see gsl_multifit_linear_genform2). Optional data weights may be supplied in the
vector w of length n, where W = diag(w).
This function computes the regularized best-fit parameters \tilde{c}
which minimize the cost function
\chi^2 = || \tilde{y} - \tilde{X} \tilde{c} ||^2 + \lambda^2 || \tilde{c} ||^2 which is
in standard form. The least squares system must therefore be converted
to standard form prior to calling this function.
The observation vector \tilde{y} is provided in ys and the matrix of
predictor variables \tilde{X} in Xs. The solution vector \tilde{c} is
returned in cs, which has length min(m,p). The SVD of Xs must be computed prior
to calling this function, using gsl_multifit_linear_svd.
The regularization parameter \lambda is provided in lambda.
The residual norm || \tilde{y} - \tilde{X} \tilde{c} || = ||y - X c||_W is returned in rnorm.
The solution norm || \tilde{c} || = ||L c|| is returned in
snorm.
After a regularized system has been solved with L = diag(\l_0,\l_1,...,\l_{p-1}), this function backtransforms the standard form solution vector cs to recover the solution vector of the original problem c. The diagonal matrix elements l_i are provided in the vector L. It is allowed to have c = cs for an in-place transform.
After a regularized system has been solved with a general rectangular matrix L,
specified by (LQR,Ltau), this function backtransforms the standard form solution cs
to recover the solution vector of the original problem, which is stored in c,
of length p. The original least squares matrix and observation vector are provided in
X and y respectively. M is the matrix computed by
gsl_multifit_linear_stdform2. For weighted fits, the weight vector
w must also be supplied.
For weighted least squares systems with L = I, this function may be used to convert the system to standard form by applying the weight matrix W = diag(w) to the least squares matrix X and observation vector y. On output, WX is equal to W^{1/2} X and Wy is equal to W^{1/2} y. It is allowed for WX = X and Wy = y for an in-place transform.
This function computes the L-curve for a least squares system
using the right hand side vector y and the SVD decomposition
of the least squares matrix X, which must be provided
to gsl_multifit_linear_svd prior to
calling this function. The output vectors reg_param,
rho, and eta must all be the same size, and will
contain the regularization parameters \lambda_i, residual norms
||y - X c_i||, and solution norms || L c_i ||
which compose the L-curve, where c_i is the regularized
solution vector corresponding to \lambda_i.
The user may determine the number of points on the L-curve by
adjusting the size of these input arrays. The regularization
parameters \lambda_i are estimated from the singular values
of X, and chosen to represent the most relevant portion of
the L-curve.
This function attempts to locate the corner of the L-curve
(||y - X c||, ||L c||) defined by the rho and eta
input arrays respectively. The corner is defined as the point of maximum
curvature of the L-curve in log-log scale. The rho and eta
arrays can be outputs of gsl_multifit_linear_lcurve. The
algorithm used simply fits a circle to 3 consecutive points on the L-curve
and uses the circle’s radius to determine the curvature at
the middle point. Therefore, the input array sizes must be
\ge 3. With more points provided for the L-curve, a better
estimate of the curvature can be obtained. The array index
corresponding to maximum curvature (ie: the corner) is returned
in idx. If the input arrays contain colinear points,
this function could fail and return GSL_EINVAL.
This function attempts to locate the corner of an alternate L-curve
(\lambda^2, ||L c||^2) studied by Rezghi and Hosseini, 2009.
This alternate L-curve can provide better estimates of the
regularization parameter for smooth solution vectors. The regularization
parameters \lambda and solution norms ||L c|| are provided
in the reg_param and eta input arrays respectively. The
corner is defined as the point of maximum curvature of this
alternate L-curve in linear scale. The reg_param and eta
arrays can be outputs of gsl_multifit_linear_lcurve. The
algorithm used simply fits a circle to 3 consecutive points on the L-curve
and uses the circle’s radius to determine the curvature at
the middle point. Therefore, the input array sizes must be
\ge 3. With more points provided for the L-curve, a better
estimate of the curvature can be obtained. The array index
corresponding to maximum curvature (ie: the corner) is returned
in idx. If the input arrays contain colinear points,
this function could fail and return GSL_EINVAL.
This function performs some initialization in preparation for computing the GCV curve and its minimum. The right hand side vector is provided in y. On output, reg_param is set to a vector of regularization parameters in decreasing order and may be of any size. The vector UTy of size p is set to U^T y. The parameter delta0 is needed for subsequent steps of the GCV calculation.
This funtion calculates the GCV curve G(\lambda) and stores it in
G on output, which must be the same size as reg_param. The
inputs reg_param, UTy and delta0 are computed in
gsl_multifit_linear_gcv_init.
This function computes the value of the regularization parameter
which minimizes the GCV curve G(\lambda) and stores it in
lambda. The input G is calculated by
gsl_multifit_linear_gcv_curve and the inputs
reg_param, UTy and delta0 are computed by
gsl_multifit_linear_gcv_init.
This function returns the value of the GCV curve G(\lambda) corresponding to the input lambda.
This function combines the steps gcv_init, gcv_curve,
and gcv_min defined above into a single function. The input
y is the right hand side vector. On output, reg_param and
G, which must be the same size, are set to vectors of
\lambda and G(\lambda) values respectively. The
output lambda is set to the optimal value of \lambda
which minimizes the GCV curve. The minimum value of the GCV curve is
returned in G_lambda.
This function computes the discrete approximation to the derivative operator L_k of order k on a regular grid of p points and stores it in L. The dimensions of L are (p-k)-by-p.
This function computes the regularization matrix L corresponding to the weighted Sobolov norm ||L c||^2 = \sum_k \alpha_k^2 ||L_k c||^2 where L_k approximates the derivative operator of order k. This regularization norm can be useful in applications where it is necessary to smooth several derivatives of the solution. p is the number of model parameters, kmax is the highest derivative to include in the summation above, and alpha is the vector of weights of size kmax + 1, where alpha[k] = \alpha_k is the weight assigned to the derivative of order k. The output matrix L is size p-by-p and upper triangular.
This function returns the reciprocal condition number of the least squares matrix X,
defined as the ratio of the smallest and largest singular values, rcond = \sigma_{min}/\sigma_{max}.
The routine gsl_multifit_linear_svd must first be called to compute the SVD of X.
Next: Robust linear regression, Previous: Multi-parameter regression, Up: Least-Squares Fitting [Index]
Next: Blocks, Up: Vectors and Matrices [Index]
All the functions are available for each of the standard data-types.
The versions for double have the prefix gsl_block,
gsl_vector and gsl_matrix. Similarly the versions for
single-precision float arrays have the prefix
gsl_block_float, gsl_vector_float and
gsl_matrix_float. The full list of available types is given
below,
gsl_block double gsl_block_float float gsl_block_long_double long double gsl_block_int int gsl_block_uint unsigned int gsl_block_long long gsl_block_ulong unsigned long gsl_block_short short gsl_block_ushort unsigned short gsl_block_char char gsl_block_uchar unsigned char gsl_block_complex complex double gsl_block_complex_float complex float gsl_block_complex_long_double complex long double
Corresponding types exist for the gsl_vector and
gsl_matrix functions.
Next: Monte Carlo Integration, Previous: Histograms, Up: Top [Index]
This chapter describes functions for creating and manipulating ntuples, sets of values associated with events. The ntuples are stored in files. Their values can be extracted in any combination and booked in a histogram using a selection function.
The values to be stored are held in a user-defined data structure, and an ntuple is created associating this data structure with a file. The values are then written to the file (normally inside a loop) using the ntuple functions described below.
A histogram can be created from ntuple data by providing a selection function and a value function. The selection function specifies whether an event should be included in the subset to be analyzed or not. The value function computes the entry to be added to the histogram for each event.
All the ntuple functions are defined in the header file gsl_ntuple.h
Next: Reading and writing vectors, Previous: Accessing vector elements, Up: Vectors [Index]
This function sets all the elements of the vector v to the value x.
This function sets all the elements of the vector v to zero.
This function makes a basis vector by setting all the elements of the vector v to zero except for the i-th element which is set to one.
Next: Real Nonsymmetric Matrices, Previous: Real Symmetric Matrices, Up: Eigensystems [Index]
For hermitian matrices, the library uses the complex form of the symmetric bidiagonalization and QR reduction method.
This function allocates a workspace for computing eigenvalues of n-by-n complex hermitian matrices. The size of the workspace is O(3n).
This function frees the memory associated with the workspace w.
This function computes the eigenvalues of the complex hermitian matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The imaginary parts of the diagonal are assumed to be zero and are not referenced. The eigenvalues are stored in the vector eval and are unordered.
This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n complex hermitian matrices. The size of the workspace is O(5n).
This function frees the memory associated with the workspace w.
This function computes the eigenvalues and eigenvectors of the complex hermitian matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The imaginary parts of the diagonal are assumed to be zero and are not referenced. The eigenvalues are stored in the vector eval and are unordered. The corresponding complex eigenvectors are stored in the columns of the matrix evec. For example, the eigenvector in the first column corresponds to the first eigenvalue. The eigenvectors are guaranteed to be mutually orthogonal and normalised to unit magnitude.
Next: Real Nonsymmetric Matrices, Previous: Real Symmetric Matrices, Up: Eigensystems [Index]
Previous: Example programs for B-splines, Up: Basis Splines [Index]
Further information on the algorithms described in this section can be found in the following book,
Further information of Greville abscissae and B-spline collocation can be found in the following paper,
A large collection of B-spline routines is available in the PPPACK library available at http://www.netlib.org/pppack, which is also part of SLATEC.
gsl-ref-html-2.3/Nonlinear-Least_002dSquares-TRS-Steihaug_002dToint-Conjugate-Gradient.html 0000664 0001750 0001750 00000010650 13055414615 027343 0 ustar edd eddOne difficulty of the dogleg methods is calculating the Gauss-Newton step when the Jacobian matrix is singular. The Steihaug-Toint method also computes a generalized dogleg step, but avoids solving for the Gauss-Newton step directly, instead using an iterative conjugate gradient algorithm. This method performs well at points where the Jacobian is singular, and is also suitable for large-scale problems where factoring the Jacobian matrix could be prohibitively expensive.
gsl-ref-html-2.3/BLAS-Examples.html 0000664 0001750 0001750 00000011663 13055414566 015136 0 ustar edd eddNext: BLAS References and Further Reading, Previous: GSL BLAS Interface, Up: BLAS Support [Index]
The following program computes the product of two matrices using the Level-3 BLAS function DGEMM,
[ 0.11 0.12 0.13 ] [ 1011 1012 ] [ 367.76 368.12 ]
[ 0.21 0.22 0.23 ] [ 1021 1022 ] = [ 674.06 674.72 ]
[ 1031 1032 ]
The matrices are stored in row major order, according to the C convention for arrays.
#include <stdio.h>
#include <gsl/gsl_blas.h>
int
main (void)
{
double a[] = { 0.11, 0.12, 0.13,
0.21, 0.22, 0.23 };
double b[] = { 1011, 1012,
1021, 1022,
1031, 1032 };
double c[] = { 0.00, 0.00,
0.00, 0.00 };
gsl_matrix_view A = gsl_matrix_view_array(a, 2, 3);
gsl_matrix_view B = gsl_matrix_view_array(b, 3, 2);
gsl_matrix_view C = gsl_matrix_view_array(c, 2, 2);
/* Compute C = A B */
gsl_blas_dgemm (CblasNoTrans, CblasNoTrans,
1.0, &A.matrix, &B.matrix,
0.0, &C.matrix);
printf ("[ %g, %g\n", c[0], c[1]);
printf (" %g, %g ]\n", c[2], c[3]);
return 0;
}
Here is the output from the program,
$ ./a.out
[ 367.76, 368.12 674.06, 674.72 ]
Next: Polynomials, Previous: Mathematical Functions, Up: Top [Index]
The functions described in this chapter provide support for complex numbers. The algorithms take care to avoid unnecessary intermediate underflows and overflows, allowing the functions to be evaluated over as much of the complex plane as possible.
For multiple-valued functions the branch cuts have been chosen to follow the conventions of Abramowitz and Stegun in the Handbook of Mathematical Functions. The functions return principal values which are the same as those in GNU Calc, which in turn are the same as those in Common Lisp, The Language (Second Edition)7 and the HP-28/48 series of calculators.
The complex types are defined in the header file gsl_complex.h, while the corresponding complex functions and arithmetic operations are defined in gsl_complex_math.h.
Next: Initializing vector elements, Previous: Vector allocation, Up: Vectors [Index]
Unlike FORTRAN compilers, C compilers do not usually provide
support for range checking of vectors and matrices.8 The functions gsl_vector_get and
gsl_vector_set can perform portable range checking for you and
report an error if you attempt to access elements outside the allowed
range.
The functions for accessing the elements of a vector or matrix are
defined in gsl_vector.h and declared extern inline to
eliminate function-call overhead. You must compile your program with
the preprocessor macro HAVE_INLINE defined to use these
functions.
If necessary you can turn off range checking completely without
modifying any source files by recompiling your program with the
preprocessor definition GSL_RANGE_CHECK_OFF. Provided your
compiler supports inline functions the effect of turning off range
checking is to replace calls to gsl_vector_get(v,i) by
v->data[i*v->stride] and calls to gsl_vector_set(v,i,x) by
v->data[i*v->stride]=x. Thus there should be no performance
penalty for using the range checking functions when range checking is
turned off.
If you use a C99 compiler which requires inline functions in header
files to be declared inline instead of extern inline,
define the macro GSL_C99_INLINE (see Inline functions).
With GCC this is selected automatically when compiling in C99 mode
(-std=c99).
If inline functions are not used, calls to the functions
gsl_vector_get and gsl_vector_set will link to the
compiled versions of these functions in the library itself. The range
checking in these functions is controlled by the global integer
variable gsl_check_range. It is enabled by default—to
disable it, set gsl_check_range to zero. Due to function-call
overhead, there is less benefit in disabling range checking here than
for inline functions.
This function returns the i-th element of a vector v. If
i lies outside the allowed range of 0 to n-1 then the error
handler is invoked and 0 is returned. An inline version of this function is used when HAVE_INLINE is defined.
This function sets the value of the i-th element of a vector
v to x. If i lies outside the allowed range of 0 to
n-1 then the error handler is invoked. An inline version of this function is used when HAVE_INLINE is defined.
These functions return a pointer to the i-th element of a vector
v. If i lies outside the allowed range of 0 to n-1
then the error handler is invoked and a null pointer is returned. Inline versions of these functions are used when HAVE_INLINE is defined.
Range
checking is available in the GNU C Compiler bounds-checking extension,
but it is not part of the default installation of GCC. Memory accesses
can also be checked with Valgrind or the gcc -fmudflap
memory protection option.
Next: Initializing vector elements, Previous: Vector allocation, Up: Vectors [Index]
Next: Elliptic Functions (Jacobi), Previous: Elementary Operations, Up: Special Functions [Index]
The functions described in this section are declared in the header file gsl_sf_ellint.h. Further information about the elliptic integrals can be found in Abramowitz & Stegun, Chapter 17.
| • Definition of Legendre Forms: | ||
| • Definition of Carlson Forms: | ||
| • Legendre Form of Complete Elliptic Integrals: | ||
| • Legendre Form of Incomplete Elliptic Integrals: | ||
| • Carlson Forms: |
Previous: Inverse Complex Hyperbolic Functions, Up: Complex Numbers [Index]
The implementations of the elementary and trigonometric functions are based on the following papers,
The general formulas and details of branch cuts can be found in the following books,
Next: Numerical integration References and Further Reading, Previous: Numerical integration error codes, Up: Numerical Integration [Index]
The integrator QAGS will handle a large class of definite
integrals. For example, consider the following integral, which has an
algebraic-logarithmic singularity at the origin,
\int_0^1 x^{-1/2} log(x) dx = -4
The program below computes this integral to a relative accuracy bound of
1e-7.
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_integration.h>
double f (double x, void * params) {
double alpha = *(double *) params;
double f = log(alpha*x) / sqrt(x);
return f;
}
int
main (void)
{
gsl_integration_workspace * w
= gsl_integration_workspace_alloc (1000);
double result, error;
double expected = -4.0;
double alpha = 1.0;
gsl_function F;
F.function = &f;
F.params = α
gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000,
w, &result, &error);
printf ("result = % .18f\n", result);
printf ("exact result = % .18f\n", expected);
printf ("estimated error = % .18f\n", error);
printf ("actual error = % .18f\n", result - expected);
printf ("intervals = %zu\n", w->size);
gsl_integration_workspace_free (w);
return 0;
}
The results below show that the desired accuracy is achieved after 8 subdivisions.
$ ./a.out
result = -4.000000000000085265 exact result = -4.000000000000000000 estimated error = 0.000000000000135447 actual error = -0.000000000000085265 intervals = 8
In fact, the extrapolation procedure used by QAGS produces an
accuracy of almost twice as many digits. The error estimate returned by
the extrapolation procedure is larger than the actual error, giving a
margin of safety of one order of magnitude.
Next: Numerical integration examples, Previous: Fixed order Gauss-Legendre integration, Up: Numerical Integration [Index]
In addition to the standard error codes for invalid arguments the functions can return the following values,
GSL_EMAXITERthe maximum number of subdivisions was exceeded.
GSL_EROUNDcannot reach tolerance because of roundoff error, or roundoff error was detected in the extrapolation table.
GSL_ESINGa non-integrable singularity or other bad integrand behavior was found in the integration interval.
GSL_EDIVERGEthe integral is divergent, or too slowly convergent to be integrated numerically.
Previous: Linking programs with the library, Up: Compiling and Linking [Index]
The following command line shows how you would link the same application with an alternative CBLAS library libcblas.a,
$ gcc example.o -lgsl -lcblas -lm
For the best performance an optimized platform-specific CBLAS
library should be used for -lcblas. The library must conform to
the CBLAS standard. The ATLAS package provides a portable
high-performance BLAS library with a CBLAS interface. It is
free software and should be installed for any work requiring fast vector
and matrix operations. The following command line will link with the
ATLAS library and its CBLAS interface,
$ gcc example.o -lgsl -lcblas -latlas -lm
If the ATLAS library is installed in a non-standard directory use
the -L option to add it to the search path, as described above.
For more information about BLAS functions see BLAS Support.
gsl-ref-html-2.3/DWT-in-one-dimension.html 0000664 0001750 0001750 00000013720 13055414550 016432 0 ustar edd eddNext: DWT in two dimension, Up: DWT Transform Functions [Index]
These functions compute in-place forward and inverse discrete wavelet
transforms of length n with stride stride on the array
data. The length of the transform n is restricted to powers
of two. For the transform version of the function the argument
dir can be either forward (+1) or backward
(-1). A workspace work of length n must be provided.
For the forward transform, the elements of the original array are replaced by the discrete wavelet transform f_i -> w_{j,k} in a packed triangular storage layout, where j is the index of the level j = 0 ... J-1 and k is the index of the coefficient within each level, k = 0 ... (2^j)-1. The total number of levels is J = \log_2(n). The output data has the following form,
(s_{-1,0}, d_{0,0}, d_{1,0}, d_{1,1}, d_{2,0}, ...,
d_{j,k}, ..., d_{J-1,2^{J-1}-1})
where the first element is the smoothing coefficient s_{-1,0}, followed by the detail coefficients d_{j,k} for each level j. The backward transform inverts these coefficients to obtain the original data.
These functions return a status of GSL_SUCCESS upon successful
completion. GSL_EINVAL is returned if n is not an integer
power of 2 or if insufficient workspace is provided.
Next: Shared Libraries, Previous: An Example Program, Up: Using the library [Index]
The library header files are installed in their own gsl directory. You should write any preprocessor include statements with a gsl/ directory prefix thus,
#include <gsl/gsl_math.h>
If the directory is not installed on the standard search path of your
compiler you will also need to provide its location to the preprocessor
as a command line flag. The default location of the gsl
directory is /usr/local/include/gsl. A typical compilation
command for a source file example.c with the GNU C compiler
gcc is,
$ gcc -Wall -I/usr/local/include -c example.c
This results in an object file example.o. The default
include path for gcc searches /usr/local/include automatically so
the -I option can actually be omitted when GSL is installed
in its default location.
| • Linking programs with the library: | ||
| • Linking with an alternative BLAS library: |
Next: Autoconf Macros, Previous: Debugging Numerical Programs, Up: Top [Index]
(See the AUTHORS file in the distribution for up-to-date information.)
Conceived GSL (with James Theiler) and wrote the design document. Wrote the simulated annealing package and the relevant chapter in the manual.
Conceived GSL (with Mark Galassi). Wrote the random number generators and the relevant chapter in this manual.
Wrote the statistical routines and the relevant chapter in this manual.
FFTs, numerical integration, random number generators and distributions, root finding, minimization and fitting, polynomial solvers, complex numbers, physical constants, permutations, vector and matrix functions, histograms, statistics, ieee-utils, revised CBLAS Level 2 & 3, matrix decompositions, eigensystems, cumulative distribution functions, testing, documentation and releases.
Wrote and documented the initial version of the root finding routines while at Los Alamos National Laboratory, Mathematical Modeling and Analysis Group.
Special Functions, Series acceleration, ODEs, BLAS, Linear Algebra, Eigensystems, Hankel Transforms.
Wrote the Monte Carlo library.
Wrote the initial complex arithmetic functions.
Wrote the initial heapsort routines and Cholesky decomposition.
Multidimensional minimization.
Implementation of the random number generators in Knuth’s Seminumerical Algorithms, 3rd Ed.
Wrote the routines for generating combinations.
Wrote the cyclic functions and the initial functions for canonical permutations.
Wrote the major cumulative distribution functions.
Wrote the routines for wavelet transforms.
Improved the implementation of the ODE solvers and wrote the ode-initval2 routines.
Implementation of the Mathieu functions.
Implementation of nonsymmetric and generalized eigensystems, B-splines, robust linear regression, and sparse matrices.
Wrote the multiset routines.
Wrote the fixed order Gauss-Legendre quadrature routines.
Wrote the CQUAD integration routines.
Thanks to Nigel Lowry for help in proofreading the manual.
The non-symmetric eigensystems routines contain code based on the LAPACK linear algebra library. LAPACK is distributed under the following license:
Copyright (c) 1992-2006 The University of Tennessee. All rights reserved.
Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:
• Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
• Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer listed in this license in the documentation and/or other materials provided with the distribution.
• Neither the name of the copyright holders nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS “AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
Next: Autoconf Macros, Previous: Debugging Numerical Programs, Up: Top [Index]
Next: Multiset Examples, Previous: Multiset functions, Up: Multisets [Index]
The library provides functions for reading and writing multisets to a file as binary data or formatted text.
This function writes the elements of the multiset c to the
stream stream in binary format. The function returns
GSL_EFAILED if there was a problem writing to the file. Since the
data is written in the native binary format it may not be portable
between different architectures.
This function reads elements from the open stream stream into the
multiset c in binary format. The multiset c must be
preallocated with correct values of n and k since the
function uses the size of c to determine how many bytes to read.
The function returns GSL_EFAILED if there was a problem reading
from the file. The data is assumed to have been written in the native
binary format on the same architecture.
This function writes the elements of the multiset c
line-by-line to the stream stream using the format specifier
format, which should be suitable for a type of size_t.
In ISO C99 the type modifier z represents size_t, so
"%zu\n" is a suitable format.11 The function returns
GSL_EFAILED if there was a problem writing to the file.
This function reads formatted data from the stream stream into the
multiset c. The multiset c must be preallocated with
correct values of n and k since the function uses the size of c to
determine how many numbers to read. The function returns
GSL_EFAILED if there was a problem reading from the file.
In versions of the
GNU C library prior to the ISO C99 standard,
the type modifier Z was used instead.
Next: Multiset Examples, Previous: Multiset functions, Up: Multisets [Index]
Previous: Multimin Examples, Up: Multidimensional Minimization [Index]
The conjugate gradient and BFGS methods are described in detail in the following book,
A brief description of multidimensional minimization algorithms and more recent references can be found in,
The simplex algorithm is described in the following paper,
Next: Handling floating point exceptions, Previous: Using gdb, Up: Debugging Numerical Programs [Index]
The contents of floating point registers can be examined using the
command info float (on supported platforms).
(gdb) info float
st0: 0xc4018b895aa17a945000 Valid Normal -7.838871e+308
st1: 0x3ff9ea3f50e4d7275000 Valid Normal 0.0285946
st2: 0x3fe790c64ce27dad4800 Valid Normal 6.7415931e-08
st3: 0x3ffaa3ef0df6607d7800 Spec Normal 0.0400229
st4: 0x3c028000000000000000 Valid Normal 4.4501477e-308
st5: 0x3ffef5412c22219d9000 Zero Normal 0.9580257
st6: 0x3fff8000000000000000 Valid Normal 1
st7: 0xc4028b65a1f6d243c800 Valid Normal -1.566206e+309
fctrl: 0x0272 53 bit; NEAR; mask DENOR UNDER LOS;
fstat: 0xb9ba flags 0001; top 7; excep DENOR OVERF UNDER LOS
ftag: 0x3fff
fip: 0x08048b5c
fcs: 0x051a0023
fopoff: 0x08086820
fopsel: 0x002b
Individual registers can be examined using the variables $reg, where reg is the register name.
(gdb) p $st1 $1 = 0.02859464454261210347719
Next: Vector properties, Previous: Vector operations, Up: Vectors [Index]
The following operations are only defined for real vectors.
This function returns the maximum value in the vector v.
This function returns the minimum value in the vector v.
This function returns the minimum and maximum values in the vector v, storing them in min_out and max_out.
This function returns the index of the maximum value in the vector v. When there are several equal maximum elements then the lowest index is returned.
This function returns the index of the minimum value in the vector v. When there are several equal minimum elements then the lowest index is returned.
This function returns the indices of the minimum and maximum values in the vector v, storing them in imin and imax. When there are several equal minimum or maximum elements then the lowest indices are returned.
These routines compute the sine function \sin(x).
These routines compute the cosine function \cos(x).
These routines compute the hypotenuse function \sqrt{x^2 + y^2} avoiding overflow and underflow.
These routines compute \sinc(x) = \sin(\pi x) / (\pi x) for any value of x.
Next: Complex Argument, Up: Dilogarithm [Index]
These routines compute the dilogarithm for a real argument. In Lewin’s notation this is Li_2(x), the real part of the dilogarithm of a real x. It is defined by the integral representation Li_2(x) = - \Re \int_0^x ds \log(1-s) / s. Note that \Im(Li_2(x)) = 0 for x <= 1, and -\pi\log(x) for x > 1.
Note that Abramowitz & Stegun refer to the Spence integral S(x)=Li_2(1-x) as the dilogarithm rather than Li_2(x).
Next: Monte Carlo Examples, Previous: MISER, Up: Monte Carlo Integration [Index]
The VEGAS algorithm of Lepage is based on importance sampling. It samples points from the probability distribution described by the function |f|, so that the points are concentrated in the regions that make the largest contribution to the integral.
In general, if the Monte Carlo integral of f is sampled with points distributed according to a probability distribution described by the function g, we obtain an estimate E_g(f; N),
E_g(f; N) = E(f/g; N)
with a corresponding variance,
\Var_g(f; N) = \Var(f/g; N).
If the probability distribution is chosen as g = |f|/I(|f|) then it can be shown that the variance V_g(f; N) vanishes, and the error in the estimate will be zero. In practice it is not possible to sample from the exact distribution g for an arbitrary function, so importance sampling algorithms aim to produce efficient approximations to the desired distribution.
The VEGAS algorithm approximates the exact distribution by making a number of passes over the integration region while histogramming the function f. Each histogram is used to define a sampling distribution for the next pass. Asymptotically this procedure converges to the desired distribution. In order to avoid the number of histogram bins growing like K^d the probability distribution is approximated by a separable function: g(x_1, x_2, ...) = g_1(x_1) g_2(x_2) ... so that the number of bins required is only Kd. This is equivalent to locating the peaks of the function from the projections of the integrand onto the coordinate axes. The efficiency of VEGAS depends on the validity of this assumption. It is most efficient when the peaks of the integrand are well-localized. If an integrand can be rewritten in a form which is approximately separable this will increase the efficiency of integration with VEGAS.
VEGAS incorporates a number of additional features, and combines both stratified sampling and importance sampling. The integration region is divided into a number of “boxes”, with each box getting a fixed number of points (the goal is 2). Each box can then have a fractional number of bins, but if the ratio of bins-per-box is less than two, Vegas switches to a kind variance reduction (rather than importance sampling).
This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions. The workspace is used to maintain the state of the integration.
This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.
This routines uses the VEGAS Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses a fixed number of function calls calls, and obtains random sampling points using the random number generator r. A previously allocated workspace s must be supplied. The result of the integration is returned in result, with an estimated absolute error abserr. The result and its error estimate are based on a weighted average of independent samples. The chi-squared per degree of freedom for the weighted average is returned via the state struct component, s->chisq, and must be consistent with 1 for the weighted average to be reliable.
This function frees the memory associated with the integrator state s.
The VEGAS algorithm computes a number of independent estimates of the
integral internally, according to the iterations parameter
described below, and returns their weighted average. Random sampling of
the integrand can occasionally produce an estimate where the error is
zero, particularly if the function is constant in some regions. An
estimate with zero error causes the weighted average to break down and
must be handled separately. In the original Fortran implementations of
VEGAS the error estimate is made non-zero by substituting a small
value (typically 1e-30). The implementation in GSL differs from
this and avoids the use of an arbitrary constant—it either assigns
the value a weight which is the average weight of the preceding
estimates or discards it according to the following procedure,
The current estimate is assigned a weight which is the average weight of the preceding estimates.
The previous estimates are discarded and the weighted averaging procedure begins with the current estimate.
The estimates are averaged using the arithmetic mean, but no error is computed.
The convergence of the algorithm can be tested using the overall chi-squared value of the results, which is available from the following function:
This function returns the chi-squared per degree of freedom for the weighted estimate of the integral. The returned value should be close to 1. A value which differs significantly from 1 indicates that the values from different iterations are inconsistent. In this case the weighted error will be under-estimated, and further iterations of the algorithm are needed to obtain reliable results.
This function returns the raw (unaveraged) values of the integral result and its error sigma from the most recent iteration of the algorithm.
The VEGAS algorithm is highly configurable. Several parameters can be changed using the following two functions.
This function copies the parameters of the integrator state into the user-supplied params structure.
This function sets the integrator parameters based on values provided in the params structure.
Typically the values of the parameters are first read using
gsl_monte_vegas_params_get, the necessary changes are made to
the fields of the params structure, and the values are copied
back into the integrator state using
gsl_monte_vegas_params_set. The functions use the
gsl_monte_vegas_params structure which contains the following
fields:
The parameter alpha controls the stiffness of the rebinning
algorithm. It is typically set between one and two. A value of zero
prevents rebinning of the grid. The default value is 1.5.
The number of iterations to perform for each call to the routine. The default value is 5 iterations.
Setting this determines the stage of the calculation. Normally,
stage = 0 which begins with a new uniform grid and empty weighted
average. Calling VEGAS with stage = 1 retains the grid from the
previous run but discards the weighted average, so that one can “tune”
the grid using a relatively small number of points and then do a large
run with stage = 1 on the optimized grid. Setting stage =
2 keeps the grid and the weighted average from the previous run, but
may increase (or decrease) the number of histogram bins in the grid
depending on the number of calls available. Choosing stage = 3
enters at the main loop, so that nothing is changed, and is equivalent
to performing additional iterations in a previous call.
The possible choices are GSL_VEGAS_MODE_IMPORTANCE,
GSL_VEGAS_MODE_STRATIFIED, GSL_VEGAS_MODE_IMPORTANCE_ONLY.
This determines whether VEGAS will use importance sampling or
stratified sampling, or whether it can pick on its own. In low
dimensions VEGAS uses strict stratified sampling (more precisely,
stratified sampling is chosen if there are fewer than 2 bins per box).
These parameters set the level of information printed by VEGAS. All
information is written to the stream ostream. The default setting
of verbose is -1, which turns off all output. A
verbose value of 0 prints summary information about the
weighted average and final result, while a value of 1 also
displays the grid coordinates. A value of 2 prints information
from the rebinning procedure for each iteration.
The above fields and the chisq value can also be accessed
directly in the gsl_monte_vegas_state but such use is
deprecated.
Next: Monte Carlo Examples, Previous: MISER, Up: Monte Carlo Integration [Index]
Next: Incomplete Beta Function, Previous: Incomplete Gamma Functions, Up: Gamma and Beta Functions [Index]
These routines compute the Beta Function, B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b) subject to a and b not being negative integers.
These routines compute the logarithm of the Beta Function, \log(B(a,b)) subject to a and b not being negative integers.
Next: Root Finding Iteration, Previous: Providing the function to solve, Up: One dimensional Root-Finding [Index]
You provide either search bounds or an initial guess; this section explains how search bounds and guesses work and how function arguments control them.
A guess is simply an x value which is iterated until it is within
the desired precision of a root. It takes the form of a double.
Search bounds are the endpoints of an interval which is iterated until the length of the interval is smaller than the requested precision. The interval is defined by two values, the lower limit and the upper limit. Whether the endpoints are intended to be included in the interval or not depends on the context in which the interval is used.
gsl-ref-html-2.3/Fermi_002dDirac-Function.html 0000664 0001750 0001750 00000010440 13055414562 017142 0 ustar edd eddNext: Gamma and Beta Functions, Previous: Exponential Integrals, Up: Special Functions [Index]
The functions described in this section are declared in the header file gsl_sf_fermi_dirac.h.
| • Complete Fermi-Dirac Integrals: | ||
| • Incomplete Fermi-Dirac Integrals: |
Next: ANSI C Compliance, Previous: Compiling and Linking, Up: Using the library [Index]
To run a program linked with the shared version of the library the operating system must be able to locate the corresponding .so file at runtime. If the library cannot be found, the following error will occur:
$ ./a.out ./a.out: error while loading shared libraries: libgsl.so.0: cannot open shared object file: No such file or directory
To avoid this error, either modify the system dynamic linker
configuration5 or
define the shell variable LD_LIBRARY_PATH to include the
directory where the library is installed.
For example, in the Bourne shell (/bin/sh or /bin/bash),
the library search path can be set with the following commands:
$ LD_LIBRARY_PATH=/usr/local/lib $ export LD_LIBRARY_PATH $ ./example
In the C-shell (/bin/csh or /bin/tcsh) the equivalent
command is,
% setenv LD_LIBRARY_PATH /usr/local/lib
The standard prompt for the C-shell in the example above is the percent character ‘%’, and should not be typed as part of the command.
To save retyping these commands each session they can be placed in an individual or system-wide login file.
To compile a statically linked version of the program, use the
-static flag in gcc,
$ gcc -static example.o -lgsl -lgslcblas -lm
Next: The 2D histogram struct, Previous: Example programs for histograms, Up: Histograms [Index]
A two dimensional histogram consists of a set of bins which count the number of events falling in a given area of the (x,y) plane. The simplest way to use a two dimensional histogram is to record two-dimensional position information, n(x,y). Another possibility is to form a joint distribution by recording related variables. For example a detector might record both the position of an event (x) and the amount of energy it deposited E. These could be histogrammed as the joint distribution n(x,E).
gsl-ref-html-2.3/Sparse-Iterative-Solver-Overview.html 0000664 0001750 0001750 00000010556 13055414613 021075 0 ustar edd eddNext: Sparse Iterative Solvers Types, Up: Sparse Iterative Solvers [Index]
Many practical iterative methods of solving large n-by-n sparse linear systems involve projecting an approximate solution for x onto a subspace of {\bf R}^n. If we define a m-dimensional subspace {\cal K} as the subspace of approximations to the solution x, then m constraints must be imposed to determine the next approximation. These m constraints define another m-dimensional subspace denoted by {\cal L}. The subspace dimension m is typically chosen to be much smaller than n in order to reduce the computational effort needed to generate the next approximate solution vector. The many iterative algorithms which exist differ mainly in their choice of {\cal K} and {\cal L}.
gsl-ref-html-2.3/Matrices.html 0000664 0001750 0001750 00000023266 13055414565 014411 0 ustar edd eddNext: Vector and Matrix References and Further Reading, Previous: Vectors, Up: Vectors and Matrices [Index]
Matrices are defined by a gsl_matrix structure which describes a
generalized slice of a block. Like a vector it represents a set of
elements in an area of memory, but uses two indices instead of one.
The gsl_matrix structure contains six components, the two
dimensions of the matrix, a physical dimension, a pointer to the memory
where the elements of the matrix are stored, data, a pointer to
the block owned by the matrix block, if any, and an ownership
flag, owner. The physical dimension determines the memory layout
and can differ from the matrix dimension to allow the use of
submatrices. The gsl_matrix structure is very simple and looks
like this,
typedef struct
{
size_t size1;
size_t size2;
size_t tda;
double * data;
gsl_block * block;
int owner;
} gsl_matrix;
Matrices are stored in row-major order, meaning that each row of
elements forms a contiguous block in memory. This is the standard
“C-language ordering” of two-dimensional arrays. Note that FORTRAN
stores arrays in column-major order. The number of rows is size1.
The range of valid row indices runs from 0 to size1-1. Similarly
size2 is the number of columns. The range of valid column indices
runs from 0 to size2-1. The physical row dimension tda, or
trailing dimension, specifies the size of a row of the matrix as
laid out in memory.
For example, in the following matrix size1 is 3, size2 is 4, and tda is 8. The physical memory layout of the matrix begins in the top left hand-corner and proceeds from left to right along each row in turn.
00 01 02 03 XX XX XX XX 10 11 12 13 XX XX XX XX 20 21 22 23 XX XX XX XX
Each unused memory location is represented by “XX”. The
pointer data gives the location of the first element of the matrix
in memory. The pointer block stores the location of the memory
block in which the elements of the matrix are located (if any). If the
matrix owns this block then the owner field is set to one and the
block will be deallocated when the matrix is freed. If the matrix is
only a slice of a block owned by another object then the owner field is
zero and any underlying block will not be freed.
The functions for allocating and accessing matrices are defined in gsl_matrix.h
Next: Vector and Matrix References and Further Reading, Previous: Vectors, Up: Vectors and Matrices [Index]
Next: Thermal Energy and Power, Previous: Volume Area and Length, Up: Physical Constants [Index]
GSL_CONST_MKSA_POUND_MASSThe mass of 1 pound.
GSL_CONST_MKSA_OUNCE_MASSThe mass of 1 ounce.
GSL_CONST_MKSA_TONThe mass of 1 ton.
GSL_CONST_MKSA_METRIC_TONThe mass of 1 metric ton (1000 kg).
GSL_CONST_MKSA_UK_TONThe mass of 1 UK ton.
GSL_CONST_MKSA_TROY_OUNCEThe mass of 1 troy ounce.
GSL_CONST_MKSA_CARATThe mass of 1 carat.
GSL_CONST_MKSA_GRAM_FORCEThe force of 1 gram weight.
GSL_CONST_MKSA_POUND_FORCEThe force of 1 pound weight.
GSL_CONST_MKSA_KILOPOUND_FORCEThe force of 1 kilopound weight.
GSL_CONST_MKSA_POUNDALThe force of 1 poundal.
Next: Arctangent Integral, Previous: Ei_3(x), Up: Exponential Integrals [Index]
These routines compute the Sine integral Si(x) = \int_0^x dt \sin(t)/t.
These routines compute the Cosine integral Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0.
Next: Zeta Functions, Previous: Transport Functions, Up: Special Functions [Index]
The library includes its own trigonometric functions in order to provide consistency across platforms and reliable error estimates. These functions are declared in the header file gsl_sf_trig.h.
Previous: Sparse BLAS operations, Up: Sparse BLAS Support [Index]
The algorithms used by these functions are described in the following sources:
Next: Carlson Forms, Previous: Legendre Form of Complete Elliptic Integrals, Up: Elliptic Integrals [Index]
These routines compute the incomplete elliptic integral F(\phi,k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.
These routines compute the incomplete elliptic integral E(\phi,k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.
These routines compute the incomplete elliptic integral \Pi(\phi,k,n) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of sign n \to -n.
These functions compute the incomplete elliptic integral D(\phi,k) which is defined through the Carlson form RD(x,y,z) by the following relation,
D(\phi,k) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).
Next: Mixed-radix FFT routines for complex data, Previous: Overview of complex data FFTs, Up: Fast Fourier Transforms [Index]
The radix-2 algorithms described in this section are simple and compact, although not necessarily the most efficient. They use the Cooley-Tukey algorithm to compute in-place complex FFTs for lengths which are a power of 2—no additional storage is required. The corresponding self-sorting mixed-radix routines offer better performance at the expense of requiring additional working space.
All the functions described in this section are declared in the header file gsl_fft_complex.h.
These functions compute forward, backward and inverse FFTs of length
n with stride stride, on the packed complex array data
using an in-place radix-2 decimation-in-time algorithm. The length of
the transform n is restricted to powers of two. For the
transform version of the function the sign argument can be
either forward (-1) or backward (+1).
The functions return a value of GSL_SUCCESS if no errors were
detected, or GSL_EDOM if the length of the data n is not a
power of two.
These are decimation-in-frequency versions of the radix-2 FFT functions.
Here is an example program which computes the FFT of a short pulse in a sample of length 128. To make the resulting Fourier transform real the pulse is defined for equal positive and negative times (-10 … 10), where the negative times wrap around the end of the array.
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>
#define REAL(z,i) ((z)[2*(i)])
#define IMAG(z,i) ((z)[2*(i)+1])
int
main (void)
{
int i; double data[2*128];
for (i = 0; i < 128; i++)
{
REAL(data,i) = 0.0; IMAG(data,i) = 0.0;
}
REAL(data,0) = 1.0;
for (i = 1; i <= 10; i++)
{
REAL(data,i) = REAL(data,128-i) = 1.0;
}
for (i = 0; i < 128; i++)
{
printf ("%d %e %e\n", i,
REAL(data,i), IMAG(data,i));
}
printf ("\n");
gsl_fft_complex_radix2_forward (data, 1, 128);
for (i = 0; i < 128; i++)
{
printf ("%d %e %e\n", i,
REAL(data,i)/sqrt(128),
IMAG(data,i)/sqrt(128));
}
return 0;
}
Note that we have assumed that the program is using the default error
handler (which calls abort for any errors). If you are not using
a safe error handler you would need to check the return status of
gsl_fft_complex_radix2_forward.
The transformed data is rescaled by 1/\sqrt n so that it fits on the same plot as the input. Only the real part is shown, by the choice of the input data the imaginary part is zero. Allowing for the wrap-around of negative times at t=128, and working in units of k/n, the DFT approximates the continuum Fourier transform, giving a modulated sine function.
Next: Mixed-radix FFT routines for complex data, Previous: Overview of complex data FFTs, Up: Fast Fourier Transforms [Index]
Next: Deprecated Functions, Previous: Aliasing of arrays, Up: Using the library [Index]
The library can be used in multi-threaded programs. All the functions
are thread-safe, in the sense that they do not use static variables.
Memory is always associated with objects and not with functions. For
functions which use workspace objects as temporary storage the
workspaces should be allocated on a per-thread basis. For functions
which use table objects as read-only memory the tables can be used
by multiple threads simultaneously. Table arguments are always declared
const in function prototypes, to indicate that they may be
safely accessed by different threads.
There are a small number of static global variables which are used to control the overall behavior of the library (e.g. whether to use range-checking, the function to call on fatal error, etc). These variables are set directly by the user, so they should be initialized once at program startup and not modified by different threads.
gsl-ref-html-2.3/Spherical-Vector-Distributions.html 0000664 0001750 0001750 00000017556 13055414507 020655 0 ustar edd eddNext: The Weibull Distribution, Previous: The Pareto Distribution, Up: Random Number Distributions [Index]
The spherical distributions generate random vectors, located on a spherical surface. They can be used as random directions, for example in the steps of a random walk.
This function returns a random direction vector v = (x,y) in two dimensions. The vector is normalized such that |v|^2 = x^2 + y^2 = 1. The obvious way to do this is to take a uniform random number between 0 and 2\pi and let x and y be the sine and cosine respectively. Two trig functions would have been expensive in the old days, but with modern hardware implementations, this is sometimes the fastest way to go. This is the case for the Pentium (but not the case for the Sun Sparcstation). One can avoid the trig evaluations by choosing x and y in the interior of a unit circle (choose them at random from the interior of the enclosing square, and then reject those that are outside the unit circle), and then dividing by \sqrt{x^2 + y^2}. A much cleverer approach, attributed to von Neumann (See Knuth, v2, 3rd ed, p140, exercise 23), requires neither trig nor a square root. In this approach, u and v are chosen at random from the interior of a unit circle, and then x=(u^2-v^2)/(u^2+v^2) and y=2uv/(u^2+v^2).
This function returns a random direction vector v = (x,y,z) in three dimensions. The vector is normalized such that |v|^2 = x^2 + y^2 + z^2 = 1. The method employed is due to Robert E. Knop (CACM 13, 326 (1970)), and explained in Knuth, v2, 3rd ed, p136. It uses the surprising fact that the distribution projected along any axis is actually uniform (this is only true for 3 dimensions).
This function returns a random direction vector v = (x_1,x_2,...,x_n) in n dimensions. The vector is normalized such that |v|^2 = x_1^2 + x_2^2 + ... + x_n^2 = 1. The method uses the fact that a multivariate Gaussian distribution is spherically symmetric. Each component is generated to have a Gaussian distribution, and then the components are normalized. The method is described by Knuth, v2, 3rd ed, p135–136, and attributed to G. W. Brown, Modern Mathematics for the Engineer (1956).
Next: The Weibull Distribution, Previous: The Pareto Distribution, Up: Random Number Distributions [Index]
Next: Matrix views, Previous: Initializing matrix elements, Up: Matrices [Index]
The library provides functions for reading and writing matrices to a file as binary data or formatted text.
This function writes the elements of the matrix m to the stream
stream in binary format. The return value is 0 for success and
GSL_EFAILED if there was a problem writing to the file. Since the
data is written in the native binary format it may not be portable
between different architectures.
This function reads into the matrix m from the open stream
stream in binary format. The matrix m must be preallocated
with the correct dimensions since the function uses the size of m to
determine how many bytes to read. The return value is 0 for success and
GSL_EFAILED if there was a problem reading from the file. The
data is assumed to have been written in the native binary format on the
same architecture.
This function writes the elements of the matrix m line-by-line to
the stream stream using the format specifier format, which
should be one of the %g, %e or %f formats for
floating point numbers and %d for integers. The function returns
0 for success and GSL_EFAILED if there was a problem writing to
the file.
This function reads formatted data from the stream stream into the
matrix m. The matrix m must be preallocated with the correct
dimensions since the function uses the size of m to determine how many
numbers to read. The function returns 0 for success and
GSL_EFAILED if there was a problem reading from the file.
Next: Matrix views, Previous: Initializing matrix elements, Up: Matrices [Index]
Next: GNU Free Documentation License, Previous: GSL CBLAS Library, Up: Top [Index]
Copyright © 2007 Free Software Foundation, Inc. http://fsf.org/ Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.
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Next: GNU Free Documentation License, Previous: GSL CBLAS Library, Up: Top [Index]
Next: Complex Numbers, Previous: Error Handling, Up: Top [Index]
This chapter describes basic mathematical functions. Some of these functions are present in system libraries, but the alternative versions given here can be used as a substitute when the system functions are not available.
The functions and macros described in this chapter are defined in the header file gsl_math.h.
Next: Random Number Distributions, Previous: Random Number Generation, Up: Top [Index]
This chapter describes functions for generating quasi-random sequences in arbitrary dimensions. A quasi-random sequence progressively covers a d-dimensional space with a set of points that are uniformly distributed. Quasi-random sequences are also known as low-discrepancy sequences. The quasi-random sequence generators use an interface that is similar to the interface for random number generators, except that seeding is not required—each generator produces a single sequence.
The functions described in this section are declared in the header file gsl_qrng.h.
Next: Irregular Spherical Bessel Functions, Previous: Irregular Modified Cylindrical Bessel Functions, Up: Bessel Functions [Index]
These routines compute the regular spherical Bessel function of zeroth order, j_0(x) = \sin(x)/x.
These routines compute the regular spherical Bessel function of first order, j_1(x) = (\sin(x)/x - \cos(x))/x.
These routines compute the regular spherical Bessel function of second order, j_2(x) = ((3/x^2 - 1)\sin(x) - 3\cos(x)/x)/x.
These routines compute the regular spherical Bessel function of order l, j_l(x), for l >= 0 and x >= 0.
This routine computes the values of the regular spherical Bessel functions j_l(x) for l from 0 to lmax inclusive for lmax >= 0 and x >= 0, storing the results in the array result_array. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
This routine uses Steed’s method to compute the values of the regular spherical Bessel functions j_l(x) for l from 0 to lmax inclusive for lmax >= 0 and x >= 0, storing the results in the array result_array. The Steed/Barnett algorithm is described in Comp. Phys. Comm. 21, 297 (1981). Steed’s method is more stable than the recurrence used in the other functions but is also slower.
Next: Irregular Spherical Bessel Functions, Previous: Irregular Modified Cylindrical Bessel Functions, Up: Bessel Functions [Index]
Next: Householder solver for linear systems, Previous: Givens Rotations, Up: Linear Algebra [Index]
A Householder transformation is a rank-1 modification of the identity matrix which can be used to zero out selected elements of a vector. A Householder matrix P takes the form,
P = I - \tau v v^T
where v is a vector (called the Householder vector) and \tau = 2/(v^T v). The functions described in this section use the rank-1 structure of the Householder matrix to create and apply Householder transformations efficiently.
This function prepares a Householder transformation P = I - \tau v v^T which can be used to zero all the elements of the input vector w except the first. On output the Householder vector v is stored in w and the scalar \tau is returned. The householder vector v is normalized so that v[0] = 1, however this 1 is not stored in the output vector. Instead, w[0] is set to the first element of the transformed vector, so that if u = P w, w[0] = u[0] on output and the remainder of u is zero.
This function applies the Householder matrix P defined by the scalar tau and the vector v to the left-hand side of the matrix A. On output the result P A is stored in A.
This function applies the Householder matrix P defined by the scalar tau and the vector v to the right-hand side of the matrix A. On output the result A P is stored in A.
This function applies the Householder transformation P defined by the scalar tau and the vector v to the vector w. On output the result P w is stored in w.
Next: Householder solver for linear systems, Previous: Givens Rotations, Up: Linear Algebra [Index]
Next: Sparse Matrices Reading and Writing, Previous: Sparse Matrices Accessing Elements, Up: Sparse Matrices [Index]
Since the sparse matrix format only stores the non-zero elements, it is automatically
initialized to zero upon allocation. The function gsl_spmatrix_set_zero may
be used to re-initialize a matrix to zero after elements have been added to it.
This function sets (or resets) all the elements of the matrix m to zero.
Next: Zeros of Derivatives of Airy Functions, Previous: Derivatives of Airy Functions, Up: Airy Functions and Derivatives [Index]
These routines compute the location of the s-th zero of the Airy function Ai(x).
These routines compute the location of the s-th zero of the Airy function Bi(x).
Next: Regular Modified Cylindrical Bessel Functions, Previous: Regular Cylindrical Bessel Functions, Up: Bessel Functions [Index]
These routines compute the irregular cylindrical Bessel function of zeroth order, Y_0(x), for x>0.
These routines compute the irregular cylindrical Bessel function of first order, Y_1(x), for x>0.
These routines compute the irregular cylindrical Bessel function of order n, Y_n(x), for x>0.
This routine computes the values of the irregular cylindrical Bessel functions Y_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The domain of the function is x>0. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
Previous: Angular Mathieu Functions, Up: Mathieu Functions [Index]
These routines compute the radial j-th kind Mathieu functions Mc_n^{(j)}(q,x) and Ms_n^{(j)}(q,x) of order n.
The allowed values of j are 1 and 2. The functions for j = 3,4 can be computed as M_n^{(3)} = M_n^{(1)} + iM_n^{(2)} and M_n^{(4)} = M_n^{(1)} - iM_n^{(2)}, where M_n^{(j)} = Mc_n^{(j)} or Ms_n^{(j)}.
These routines compute a series of the radial Mathieu functions of kind j, with order from nmin to nmax inclusive, storing the results in the array result_array.
Next: Trigonometric Functions, Previous: Synchrotron Functions, Up: Special Functions [Index]
The transport functions J(n,x) are defined by the integral representations J(n,x) := \int_0^x dt t^n e^t /(e^t - 1)^2. They are declared in the header file gsl_sf_transport.h.
These routines compute the transport function J(2,x).
These routines compute the transport function J(3,x).
These routines compute the transport function J(4,x).
These routines compute the transport function J(5,x).
Next: Nonlinear Least-Squares Function Definition, Previous: Nonlinear Least-Squares Tunable Parameters, Up: Nonlinear Least-Squares Fitting [Index]
These functions return a pointer to a newly allocated instance of a
derivative solver of type T for n observations and p
parameters. The params input specifies a tunable set of
parameters which will affect important details in each iteration
of the trust region subproblem algorithm. It is recommended to start
with the suggested default parameters (see
gsl_multifit_nlinear_default_parameters and
gsl_multilarge_nlinear_default_parameters) and then tune
the parameters once the code is working correctly. See
Nonlinear Least-Squares Tunable Parameters
for descriptions of the various parameters.
For example, the following code creates an instance of a
Levenberg-Marquardt solver for 100 data points and 3 parameters,
using suggested defaults:
const gsl_multifit_nlinear_type * T
= gsl_multifit_nlinear_lm;
gsl_multifit_nlinear_parameters params
= gsl_multifit_nlinear_default_parameters();
gsl_multifit_nlinear_workspace * w
= gsl_multifit_nlinear_alloc (T, ¶ms, 100, 3);
The number of observations n must be greater than or equal to parameters p.
If there is insufficient memory to create the solver then the function
returns a null pointer and the error handler is invoked with an error
code of GSL_ENOMEM.
These functions return a set of recommended default parameters for use in solving nonlinear least squares problems. The user can tune each parameter to improve the performance on their particular problem, see Nonlinear Least-Squares Tunable Parameters.
These functions initialize, or reinitialize, an existing workspace w to use the system fdf and the initial guess x. See Nonlinear Least-Squares Function Definition for a description of the fdf structure.
Optionally, a weight vector wts can be given to perform a weighted nonlinear regression. Here, the weighting matrix is W = diag(w_1,w_2,...,w_n).
These functions free all the memory associated with the workspace w.
These functions return a pointer to the name of the solver. For example,
printf ("w is a '%s' solver\n",
gsl_multifit_nlinear_name (w));
would print something like w is a 'trust-region' solver.
These functions return a pointer to the name of the trust region subproblem method. For example,
printf ("w is a '%s' solver\n",
gsl_multifit_nlinear_trs_name (w));
would print something like w is a 'levenberg-marquardt' solver.
Next: Nonlinear Least-Squares Function Definition, Previous: Nonlinear Least-Squares Tunable Parameters, Up: Nonlinear Least-Squares Fitting [Index]
Next: Large Dense Linear Systems Solution Steps, Previous: Large Dense Linear Systems Normal Equations, Up: Large Dense Linear Systems [Index]
An algorithm which has better numerical stability for ill-conditioned problems is known as the Tall Skinny QR (TSQR) method. This method is based on computing the thin QR decomposition of the least squares matrix X = Q R, where Q is an n-by-p matrix with orthogonal columns, and R is a p-by-p upper triangular matrix. Once these factors are calculated, the residual becomes
\chi^2 = || Q^T y - R c ||^2 + \lambda^2 || c ||^2
which can be written as the matrix equation
[ R ; \lambda I ] c = [ Q^T b ; 0 ]
The matrix on the left hand side is now a much smaller 2p-by-p matrix which can be solved with a standard SVD approach. The Q matrix is just as large as the original matrix X, however it does not need to be explicitly constructed. The TSQR algorithm computes only the p-by-p matrix R and the p-by-1 vector Q^T y, and updates these quantities as new blocks are added to the system. Each time a new block of rows (X_i,y_i) is added, the algorithm performs a QR decomposition of the matrix
[ R_(i-1) ; X_i ]
where R_{i-1} is the upper triangular R factor for the matrix
[ X_1 ; ... ; X_(i-1) ]
This QR decomposition is done efficiently taking into account the sparse structure of R_{i-1}. See Demmel et al, 2008 for more details on how this is accomplished. The number of operations for this method is O(2np^2 - {2 \over 3}p^3).
Next: Large Dense Linear Systems Solution Steps, Previous: Large Dense Linear Systems Normal Equations, Up: Large Dense Linear Systems [Index]
Next: QAWF adaptive integration for Fourier integrals, Previous: QAWS adaptive integration for singular functions, Up: Numerical Integration [Index]
The QAWO algorithm is designed for integrands with an oscillatory factor, \sin(\omega x) or \cos(\omega x). In order to work efficiently the algorithm requires a table of Chebyshev moments which must be pre-computed with calls to the functions below.
This function allocates space for a gsl_integration_qawo_table
struct and its associated workspace describing a sine or cosine weight
function W(x) with the parameters (\omega, L),
W(x) = sin(omega x) W(x) = cos(omega x)
The parameter L must be the length of the interval over which the function will be integrated L = b - a. The choice of sine or cosine is made with the parameter sine which should be chosen from one of the two following symbolic values:
GSL_INTEG_COSINE GSL_INTEG_SINE
The gsl_integration_qawo_table is a table of the trigonometric
coefficients required in the integration process. The parameter n
determines the number of levels of coefficients that are computed. Each
level corresponds to one bisection of the interval L, so that
n levels are sufficient for subintervals down to the length
L/2^n. The integration routine gsl_integration_qawo
returns the error GSL_ETABLE if the number of levels is
insufficient for the requested accuracy.
This function changes the parameters omega, L and sine of the existing workspace t.
This function allows the length parameter L of the workspace t to be changed.
This function frees all the memory associated with the workspace t.
This function uses an adaptive algorithm to compute the integral of f over (a,b) with the weight function \sin(\omega x) or \cos(\omega x) defined by the table wf,
I = \int_a^b dx f(x) sin(omega x) I = \int_a^b dx f(x) cos(omega x)
The results are extrapolated using the epsilon-algorithm to accelerate the convergence of the integral. The function returns the final approximation from the extrapolation, result, and an estimate of the absolute error, abserr. The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals is given by limit, which may not exceed the allocated size of the workspace.
Those subintervals with “large” widths d where d\omega > 4 are computed using a 25-point Clenshaw-Curtis integration rule, which handles the oscillatory behavior. Subintervals with a “small” widths where d\omega < 4 are computed using a 15-point Gauss-Kronrod integration.
Next: QAWF adaptive integration for Fourier integrals, Previous: QAWS adaptive integration for singular functions, Up: Numerical Integration [Index]
Next: Sparse Matrices Exchanging Rows and Columns, Previous: Sparse Matrices Reading and Writing, Up: Sparse Matrices [Index]
This function copies the elements of the sparse matrix src into dest. The two matrices must have the same dimensions and be in the same storage format.
Next: Sparse Matrices Compressed Format, Previous: Sparse Matrices Properties, Up: Sparse Matrices [Index]
This function returns the minimum and maximum elements of the matrix m, storing them in min_out and max_out, and searching only the non-zero values.
Next: Complex Hermitian Matrices, Up: Eigensystems [Index]
For real symmetric matrices, the library uses the symmetric bidiagonalization and QR reduction method. This is described in Golub & van Loan, section 8.3. The computed eigenvalues are accurate to an absolute accuracy of \epsilon ||A||_2, where \epsilon is the machine precision.
This function allocates a workspace for computing eigenvalues of n-by-n real symmetric matrices. The size of the workspace is O(2n).
This function frees the memory associated with the workspace w.
This function computes the eigenvalues of the real symmetric matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The eigenvalues are stored in the vector eval and are unordered.
This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real symmetric matrices. The size of the workspace is O(4n).
This function frees the memory associated with the workspace w.
This function computes the eigenvalues and eigenvectors of the real symmetric matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The eigenvalues are stored in the vector eval and are unordered. The corresponding eigenvectors are stored in the columns of the matrix evec. For example, the eigenvector in the first column corresponds to the first eigenvalue. The eigenvectors are guaranteed to be mutually orthogonal and normalised to unit magnitude.
Next: Complex Hermitian Matrices, Up: Eigensystems [Index]
Next: Cubic Equations, Previous: Divided Difference Representation of Polynomials, Up: Polynomials [Index]
This function finds the real roots of the quadratic equation,
a x^2 + b x + c = 0
The number of real roots (either zero, one or two) is returned, and their locations are stored in x0 and x1. If no real roots are found then x0 and x1 are not modified. If one real root is found (i.e. if a=0) then it is stored in x0. When two real roots are found they are stored in x0 and x1 in ascending order. The case of coincident roots is not considered special. For example (x-1)^2=0 will have two roots, which happen to have exactly equal values.
The number of roots found depends on the sign of the discriminant b^2 - 4 a c. This will be subject to rounding and cancellation errors when computed in double precision, and will also be subject to errors if the coefficients of the polynomial are inexact. These errors may cause a discrete change in the number of roots. However, for polynomials with small integer coefficients the discriminant can always be computed exactly.
This function finds the complex roots of the quadratic equation,
a z^2 + b z + c = 0
The number of complex roots is returned (either one or two) and the locations of the roots are stored in z0 and z1. The roots are returned in ascending order, sorted first by their real components and then by their imaginary components. If only one real root is found (i.e. if a=0) then it is stored in z0.
Next: Cubic Equations, Previous: Divided Difference Representation of Polynomials, Up: Polynomials [Index]
Next: Conventions used in this manual, Previous: Reporting Bugs, Up: Introduction [Index]
Additional information, including online copies of this manual, links to related projects, and mailing list archives are available from the website mentioned above.
Any questions about the use and installation of the library can be asked
on the mailing list help-gsl@gnu.org. To subscribe to this
list, send an email of the following form:
To: help-gsl-request@gnu.org Subject: subscribe
This mailing list can be used to ask questions not covered by this manual, and to contact the developers of the library.
If you would like to refer to the GNU Scientific Library in a journal article, the recommended way is to cite this reference manual, e.g. M. Galassi et al, GNU Scientific Library Reference Manual (3rd Ed.), ISBN 0954612078.
If you want to give a url, use “http://www.gnu.org/software/gsl/”.
gsl-ref-html-2.3/Measurement-of-Time.html 0000664 0001750 0001750 00000007722 13055414606 016420 0 ustar edd eddNext: Imperial Units, Previous: Atomic and Nuclear Physics, Up: Physical Constants [Index]
GSL_CONST_MKSA_MINUTEThe number of seconds in 1 minute.
GSL_CONST_MKSA_HOURThe number of seconds in 1 hour.
GSL_CONST_MKSA_DAYThe number of seconds in 1 day.
GSL_CONST_MKSA_WEEKThe number of seconds in 1 week.
Next: Sorting vectors, Up: Sorting [Index]
The following function provides a simple alternative to the standard
library function qsort. It is intended for systems lacking
qsort, not as a replacement for it. The function qsort
should be used whenever possible, as it will be faster and can provide
stable ordering of equal elements. Documentation for qsort is
available in the GNU C Library Reference Manual.
The functions described in this section are defined in the header file gsl_heapsort.h.
This function sorts the count elements of the array array, each of size size, into ascending order using the comparison function compare. The type of the comparison function is defined by,
int (*gsl_comparison_fn_t) (const void * a,
const void * b)
A comparison function should return a negative integer if the first
argument is less than the second argument, 0 if the two arguments
are equal and a positive integer if the first argument is greater than
the second argument.
For example, the following function can be used to sort doubles into ascending numerical order.
int
compare_doubles (const double * a,
const double * b)
{
if (*a > *b)
return 1;
else if (*a < *b)
return -1;
else
return 0;
}
The appropriate function call to perform the sort is,
gsl_heapsort (array, count, sizeof(double),
compare_doubles);
Note that unlike qsort the heapsort algorithm cannot be made into
a stable sort by pointer arithmetic. The trick of comparing pointers for
equal elements in the comparison function does not work for the heapsort
algorithm. The heapsort algorithm performs an internal rearrangement of
the data which destroys its initial ordering.
This function indirectly sorts the count elements of the array array, each of size size, into ascending order using the comparison function compare. The resulting permutation is stored in p, an array of length n. The elements of p give the index of the array element which would have been stored in that position if the array had been sorted in place. The first element of p gives the index of the least element in array, and the last element of p gives the index of the greatest element in array. The array itself is not changed.
Next: Sorting vectors, Up: Sorting [Index]
Previous: Example of accelerating a series, Up: Series Acceleration [Index]
The algorithms used by these functions are described in the following papers,
The theory of the u-transform was presented by Levin,
A review paper on the Levin Transform is available online,
Next: Simulated Annealing functions, Up: Simulated Annealing [Index]
The simulated annealing algorithm takes random walks through the problem space, looking for points with low energies; in these random walks, the probability of taking a step is determined by the Boltzmann distribution,
p = e^{-(E_{i+1} - E_i)/(kT)}
if E_{i+1} > E_i, and p = 1 when E_{i+1} <= E_i.
In other words, a step will occur if the new energy is lower. If the new energy is higher, the transition can still occur, and its likelihood is proportional to the temperature T and inversely proportional to the energy difference E_{i+1} - E_i.
The temperature T is initially set to a high value, and a random walk is carried out at that temperature. Then the temperature is lowered very slightly according to a cooling schedule, for example: T -> T/mu_T where \mu_T is slightly greater than 1.
The slight probability of taking a step that gives higher energy is what allows simulated annealing to frequently get out of local minima.
gsl-ref-html-2.3/1D-Index-Look_002dup-and-Acceleration.html 0000664 0001750 0001750 00000015541 13055414457 021273 0 ustar edd eddNext: 1D Evaluation of Interpolating Functions, Previous: 1D Interpolation Types, Up: Interpolation [Index]
The state of searches can be stored in a gsl_interp_accel object,
which is a kind of iterator for interpolation lookups. It caches the
previous value of an index lookup. When the subsequent interpolation
point falls in the same interval its index value can be returned
immediately.
This function returns the index i of the array x_array such
that x_array[i] <= x < x_array[i+1]. The index is searched for
in the range [index_lo,index_hi]. An inline version of this function is used when HAVE_INLINE is defined.
This function returns a pointer to an accelerator object, which is a kind of iterator for interpolation lookups. It tracks the state of lookups, thus allowing for application of various acceleration strategies.
This function performs a lookup action on the data array x_array
of size size, using the given accelerator a. This is how
lookups are performed during evaluation of an interpolation. The
function returns an index i such that x_array[i] <= x <
x_array[i+1]. An inline version of this function is used when HAVE_INLINE is defined.
This function reinitializes the accelerator object acc. It should be used when the cached information is no longer applicable—for example, when switching to a new dataset.
This function frees the accelerator object acc.
Next: 1D Evaluation of Interpolating Functions, Previous: 1D Interpolation Types, Up: Interpolation [Index]
Next: Special Functions References and Further Reading, Previous: Zeta Functions, Up: Special Functions [Index]
The following example demonstrates the use of the error handling form of the special functions, in this case to compute the Bessel function J_0(5.0),
#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_bessel.h>
int
main (void)
{
double x = 5.0;
gsl_sf_result result;
double expected = -0.17759677131433830434739701;
int status = gsl_sf_bessel_J0_e (x, &result);
printf ("status = %s\n", gsl_strerror(status));
printf ("J0(5.0) = %.18f\n"
" +/- % .18f\n",
result.val, result.err);
printf ("exact = %.18f\n", expected);
return status;
}
Here are the results of running the program,
$ ./a.out
status = success
J0(5.0) = -0.177596771314338264
+/- 0.000000000000000193
exact = -0.177596771314338292
The next program computes the same quantity using the natural form of the function. In this case the error term result.err and return status are not accessible.
#include <stdio.h>
#include <gsl/gsl_sf_bessel.h>
int
main (void)
{
double x = 5.0;
double expected = -0.17759677131433830434739701;
double y = gsl_sf_bessel_J0 (x);
printf ("J0(5.0) = %.18f\n", y);
printf ("exact = %.18f\n", expected);
return 0;
}
The results of the function are the same,
$ ./a.out
J0(5.0) = -0.177596771314338264 exact = -0.177596771314338292
Next: Wavelet Transforms, Previous: Chebyshev Approximations, Up: Top [Index]
The functions described in this chapter accelerate the convergence of a series using the Levin u-transform. This method takes a small number of terms from the start of a series and uses a systematic approximation to compute an extrapolated value and an estimate of its error. The u-transform works for both convergent and divergent series, including asymptotic series.
These functions are declared in the header file gsl_sum.h.
| • Acceleration functions: | ||
| • Acceleration functions without error estimation: | ||
| • Example of accelerating a series: | ||
| • Series Acceleration References: |
Next: The Gamma Distribution, Previous: The Levy alpha-Stable Distributions, Up: Random Number Distributions [Index]
This function returns a random variate from the Levy skew stable distribution with scale c, exponent alpha and skewness parameter beta. The skewness parameter must lie in the range [-1,1]. The Levy skew stable probability distribution is defined by a Fourier transform,
p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha/2)))
When \alpha = 1 the term \tan(\pi \alpha/2) is replaced by
-(2/\pi)\log|t|. There is no explicit solution for the form of
p(x) and the library does not define a corresponding pdf
function. For \alpha = 2 the distribution reduces to a Gaussian
distribution with \sigma = \sqrt{2} c and the skewness parameter has no effect.
For \alpha < 1 the tails of the distribution become extremely
wide. The symmetric distribution corresponds to \beta =
0.
The algorithm only works for 0 < alpha <= 2.
The Levy alpha-stable distributions have the property that if N alpha-stable variates are drawn from the distribution p(c, \alpha, \beta) then the sum Y = X_1 + X_2 + \dots + X_N will also be distributed as an alpha-stable variate, p(N^(1/\alpha) c, \alpha, \beta).
Next: Weighted Samples, Previous: Covariance, Up: Statistics [Index]
This function efficiently computes the Pearson correlation coefficient between the datasets data1 and data2 which must both be of the same length n.
r = cov(x, y) / (\Hat\sigma_x \Hat\sigma_y)
= {1/(n-1) \sum (x_i - \Hat x) (y_i - \Hat y)
\over
\sqrt{1/(n-1) \sum (x_i - \Hat x)^2} \sqrt{1/(n-1) \sum (y_i - \Hat y)^2}
}
This function computes the Spearman rank correlation coefficient between the datasets data1 and data2 which must both be of the same length n. Additional workspace of size 2*n is required in work. The Spearman rank correlation between vectors x and y is equivalent to the Pearson correlation between the ranked vectors x_R and y_R, where ranks are defined to be the average of the positions of an element in the ascending order of the values.
Next: Resampling from histograms, Previous: Histogram Operations, Up: Histograms [Index]
The library provides functions for reading and writing histograms to a file as binary data or formatted text.
This function writes the ranges and bins of the histogram h to the
stream stream in binary format. The return value is 0 for success
and GSL_EFAILED if there was a problem writing to the file. Since
the data is written in the native binary format it may not be portable
between different architectures.
This function reads into the histogram h from the open stream
stream in binary format. The histogram h must be
preallocated with the correct size since the function uses the number of
bins in h to determine how many bytes to read. The return value is
0 for success and GSL_EFAILED if there was a problem reading from
the file. The data is assumed to have been written in the native binary
format on the same architecture.
This function writes the ranges and bins of the histogram h
line-by-line to the stream stream using the format specifiers
range_format and bin_format. These should be one of the
%g, %e or %f formats for floating point
numbers. The function returns 0 for success and GSL_EFAILED if
there was a problem writing to the file. The histogram output is
formatted in three columns, and the columns are separated by spaces,
like this,
range[0] range[1] bin[0] range[1] range[2] bin[1] range[2] range[3] bin[2] .... range[n-1] range[n] bin[n-1]
The values of the ranges are formatted using range_format and the value of the bins are formatted using bin_format. Each line contains the lower and upper limit of the range of the bins and the value of the bin itself. Since the upper limit of one bin is the lower limit of the next there is duplication of these values between lines but this allows the histogram to be manipulated with line-oriented tools.
This function reads formatted data from the stream stream into the
histogram h. The data is assumed to be in the three-column format
used by gsl_histogram_fprintf. The histogram h must be
preallocated with the correct length since the function uses the size of
h to determine how many numbers to read. The function returns 0
for success and GSL_EFAILED if there was a problem reading from
the file.
Next: Resampling from histograms, Previous: Histogram Operations, Up: Histograms [Index]
Next: Eigenvalue and Eigenvector Examples, Previous: Real Generalized Nonsymmetric Eigensystems, Up: Eigensystems [Index]
This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding real eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value of the parameter sort_type,
GSL_EIGEN_SORT_VAL_ASCascending order in numerical value
GSL_EIGEN_SORT_VAL_DESCdescending order in numerical value
GSL_EIGEN_SORT_ABS_ASCascending order in magnitude
GSL_EIGEN_SORT_ABS_DESCdescending order in magnitude
This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding complex eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value of the parameter sort_type as shown above.
This function simultaneously sorts the eigenvalues stored in the vector
eval and the corresponding complex eigenvectors stored in the
columns of the matrix evec into ascending or descending order
according to the value of the parameter sort_type as shown above.
Only GSL_EIGEN_SORT_ABS_ASC and GSL_EIGEN_SORT_ABS_DESC are
supported due to the eigenvalues being complex.
This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding real eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value of the parameter sort_type as shown above.
This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding complex eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value of the parameter sort_type as shown above.
This function simultaneously sorts the eigenvalues stored in the vectors
(alpha, beta) and the corresponding complex eigenvectors
stored in the columns of the matrix evec into ascending or
descending order according to the value of the parameter sort_type
as shown above. Only GSL_EIGEN_SORT_ABS_ASC and
GSL_EIGEN_SORT_ABS_DESC are supported due to the eigenvalues being
complex.
Next: Eigenvalue and Eigenvector Examples, Previous: Real Generalized Nonsymmetric Eigensystems, Up: Eigensystems [Index]
Next: Linear Algebra Examples, Previous: Triangular Systems, Up: Linear Algebra [Index]
The process of balancing a matrix applies similarity transformations to make the rows and columns have comparable norms. This is useful, for example, to reduce roundoff errors in the solution of eigenvalue problems. Balancing a matrix A consists of replacing A with a similar matrix
A' = D^(-1) A D
where D is a diagonal matrix whose entries are powers of the floating point radix.
This function replaces the matrix A with its balanced counterpart and stores the diagonal elements of the similarity transformation into the vector D.
Next: Nonlinear Least-Squares References and Further Reading, Previous: Nonlinear Least-Squares Troubleshooting, Up: Nonlinear Least-Squares Fitting [Index]
The following example programs demonstrate the nonlinear least squares fitting capabilities.
| • Nonlinear Least-Squares Exponential Fit Example: | ||
| • Nonlinear Least-Squares Geodesic Acceleration Example: | ||
| • Nonlinear Least-Squares Comparison Example: | ||
| • Nonlinear Least-Squares Large Example: |
Next: 2D Higher-level Interface, Previous: 2D Interpolation Types, Up: Interpolation [Index]
These functions return the interpolated value of z for a given
point (x,y), using the interpolation object interp, data
arrays xa, ya, and za and the accelerators xacc
and yacc. When x is outside the range of xa or y
is outside the range of ya, the error code
GSL_EDOM is returned.
These functions return the interpolated value of z for a given point (x,y), using the interpolation object interp, data arrays xa, ya, and za and the accelerators xacc and yacc. The functions perform no bounds checking, so when x is outside the range of xa or y is outside the range of ya, extrapolation is performed.
These functions return the interpolated value d
= \partial z / \partial x for a given point (x,y),
using the interpolation object interp, data
arrays xa, ya, and za and the accelerators xacc
and yacc. When x is outside the range of xa or y
is outside the range of ya, the error code
GSL_EDOM is returned.
These functions return the interpolated value d
= \partial z / \partial y for a given point (x,y),
using the interpolation object interp, data
arrays xa, ya, and za and the accelerators xacc
and yacc. When x is outside the range of xa or y
is outside the range of ya, the error code
GSL_EDOM is returned.
These functions return the interpolated value d
= \partial^2 z / \partial x^2 for a given point (x,y),
using the interpolation object interp, data
arrays xa, ya, and za and the accelerators xacc
and yacc. When x is outside the range of xa or y
is outside the range of ya, the error code
GSL_EDOM is returned.
These functions return the interpolated value d
= \partial^2 z / \partial y^2 for a given point (x,y),
using the interpolation object interp, data
arrays xa, ya, and za and the accelerators xacc
and yacc. When x is outside the range of xa or y
is outside the range of ya, the error code
GSL_EDOM is returned.
These functions return the interpolated value d
= \partial^2 z / \partial x \partial y for a given point (x,y),
using the interpolation object interp, data
arrays xa, ya, and za and the accelerators xacc
and yacc. When x is outside the range of xa or y
is outside the range of ya, the error code
GSL_EDOM is returned.
Next: 2D Higher-level Interface, Previous: 2D Interpolation Types, Up: Interpolation [Index]
Next: Statistics References and Further Reading, Previous: Median and Percentiles, Up: Statistics [Index]
Here is a basic example of how to use the statistical functions:
#include <stdio.h>
#include <gsl/gsl_statistics.h>
int
main(void)
{
double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
double mean, variance, largest, smallest;
mean = gsl_stats_mean(data, 1, 5);
variance = gsl_stats_variance(data, 1, 5);
largest = gsl_stats_max(data, 1, 5);
smallest = gsl_stats_min(data, 1, 5);
printf ("The dataset is %g, %g, %g, %g, %g\n",
data[0], data[1], data[2], data[3], data[4]);
printf ("The sample mean is %g\n", mean);
printf ("The estimated variance is %g\n", variance);
printf ("The largest value is %g\n", largest);
printf ("The smallest value is %g\n", smallest);
return 0;
}
The program should produce the following output,
The dataset is 17.2, 18.1, 16.5, 18.3, 12.6 The sample mean is 16.54 The estimated variance is 5.373 The largest value is 18.3 The smallest value is 12.6
Here is an example using sorted data,
#include <stdio.h>
#include <gsl/gsl_sort.h>
#include <gsl/gsl_statistics.h>
int
main(void)
{
double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
double median, upperq, lowerq;
printf ("Original dataset: %g, %g, %g, %g, %g\n",
data[0], data[1], data[2], data[3], data[4]);
gsl_sort (data, 1, 5);
printf ("Sorted dataset: %g, %g, %g, %g, %g\n",
data[0], data[1], data[2], data[3], data[4]);
median
= gsl_stats_median_from_sorted_data (data,
1, 5);
upperq
= gsl_stats_quantile_from_sorted_data (data,
1, 5,
0.75);
lowerq
= gsl_stats_quantile_from_sorted_data (data,
1, 5,
0.25);
printf ("The median is %g\n", median);
printf ("The upper quartile is %g\n", upperq);
printf ("The lower quartile is %g\n", lowerq);
return 0;
}
This program should produce the following output,
Original dataset: 17.2, 18.1, 16.5, 18.3, 12.6 Sorted dataset: 12.6, 16.5, 17.2, 18.1, 18.3 The median is 17.2 The upper quartile is 18.1 The lower quartile is 16.5
Next: Statistics References and Further Reading, Previous: Median and Percentiles, Up: Statistics [Index]
Next: 1D Interpolation Example programs, Previous: 1D Evaluation of Interpolating Functions, Up: Interpolation [Index]
The functions described in the previous sections required the user to
supply pointers to the x and y arrays on each call. The
following functions are equivalent to the corresponding
gsl_interp functions but maintain a copy of this data in the
gsl_spline object. This removes the need to pass both xa
and ya as arguments on each evaluation. These functions are
defined in the header file gsl_spline.h.
Next: GSL is Free Software, Up: Introduction [Index]
The library covers a wide range of topics in numerical computing. Routines are available for the following areas,
| Complex Numbers | Roots of Polynomials | |
| Special Functions | Vectors and Matrices | |
| Permutations | Combinations | |
| Sorting | BLAS Support | |
| Linear Algebra | CBLAS Library | |
| Fast Fourier Transforms | Eigensystems | |
| Random Numbers | Quadrature | |
| Random Distributions | Quasi-Random Sequences | |
| Histograms | Statistics | |
| Monte Carlo Integration | N-Tuples | |
| Differential Equations | Simulated Annealing | |
| Numerical Differentiation | Interpolation | |
| Series Acceleration | Chebyshev Approximations | |
| Root-Finding | Discrete Hankel Transforms | |
| Least-Squares Fitting | Minimization | |
| IEEE Floating-Point | Physical Constants | |
| Basis Splines | Wavelets |
The use of these routines is described in this manual. Each chapter provides detailed definitions of the functions, followed by example programs and references to the articles on which the algorithms are based.
Where possible the routines have been based on reliable public-domain packages such as FFTPACK and QUADPACK, which the developers of GSL have reimplemented in C with modern coding conventions.
gsl-ref-html-2.3/Sorting.html 0000664 0001750 0001750 00000012554 13055414417 014261 0 ustar edd eddNext: BLAS Support, Previous: Multisets, Up: Top [Index]
This chapter describes functions for sorting data, both directly and indirectly (using an index). All the functions use the heapsort algorithm. Heapsort is an O(N \log N) algorithm which operates in-place and does not require any additional storage. It also provides consistent performance, the running time for its worst-case (ordered data) being not significantly longer than the average and best cases. Note that the heapsort algorithm does not preserve the relative ordering of equal elements—it is an unstable sort. However the resulting order of equal elements will be consistent across different platforms when using these functions.
| • Sorting objects: | ||
| • Sorting vectors: | ||
| • Selecting the k smallest or largest elements: | ||
| • Computing the rank: | ||
| • Sorting Examples: | ||
| • Sorting References and Further Reading: |
Next: Finding maximum and minimum elements of vectors, Previous: Exchanging elements, Up: Vectors [Index]
This function adds the elements of vector b to the elements of vector a. The result a_i \leftarrow a_i + b_i is stored in a and b remains unchanged. The two vectors must have the same length.
This function subtracts the elements of vector b from the elements of vector a. The result a_i \leftarrow a_i - b_i is stored in a and b remains unchanged. The two vectors must have the same length.
This function multiplies the elements of vector a by the elements of vector b. The result a_i \leftarrow a_i * b_i is stored in a and b remains unchanged. The two vectors must have the same length.
This function divides the elements of vector a by the elements of vector b. The result a_i \leftarrow a_i / b_i is stored in a and b remains unchanged. The two vectors must have the same length.
This function multiplies the elements of vector a by the constant factor x. The result a_i \leftarrow x a_i is stored in a.
This function adds the constant value x to the elements of the vector a. The result a_i \leftarrow a_i + x is stored in a.
Next: Factorials, Up: Gamma and Beta Functions [Index]
The Gamma function is defined by the following integral,
\Gamma(x) = \int_0^\infty dt t^{x-1} \exp(-t)
It is related to the factorial function by \Gamma(n)=(n-1)! for positive integer n. Further information on the Gamma function can be found in Abramowitz & Stegun, Chapter 6.
These routines compute the Gamma function \Gamma(x), subject to x
not being a negative integer or zero. The function is computed using the real
Lanczos method. The maximum value of x such that \Gamma(x) is not
considered an overflow is given by the macro GSL_SF_GAMMA_XMAX
and is 171.0.
These routines compute the logarithm of the Gamma function, \log(\Gamma(x)), subject to x not being a negative integer or zero. For x<0 the real part of \log(\Gamma(x)) is returned, which is equivalent to \log(|\Gamma(x)|). The function is computed using the real Lanczos method.
This routine computes the sign of the gamma function and the logarithm of its magnitude, subject to x not being a negative integer or zero. The function is computed using the real Lanczos method. The value of the gamma function and its error can be reconstructed using the relation \Gamma(x) = sgn * \exp(result\_lg), taking into account the two components of result_lg.
These routines compute the regulated Gamma Function \Gamma^*(x) for x > 0. The regulated gamma function is given by,
\Gamma^*(x) = \Gamma(x)/(\sqrt{2\pi} x^{(x-1/2)} \exp(-x))
= (1 + (1/12x) + ...) for x \to \infty
and is a useful suggestion of Temme.
These routines compute the reciprocal of the gamma function, 1/\Gamma(x) using the real Lanczos method.
This routine computes \log(\Gamma(z)) for complex z=z_r+i
z_i and z not a negative integer or zero, using the complex Lanczos
method. The returned parameters are lnr = \log|\Gamma(z)| and
arg = \arg(\Gamma(z)) in (-\pi,\pi]. Note that the phase
part (arg) is not well-determined when |z| is very large,
due to inevitable roundoff in restricting to (-\pi,\pi]. This
will result in a GSL_ELOSS error when it occurs. The absolute
value part (lnr), however, never suffers from loss of precision.
Next: Factorials, Up: Gamma and Beta Functions [Index]
Next: Eigenvalue and Eigenvector References, Previous: Sorting Eigenvalues and Eigenvectors, Up: Eigensystems [Index]
The following program computes the eigenvalues and eigenvectors of the 4-th order Hilbert matrix, H(i,j) = 1/(i + j + 1).
#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_eigen.h>
int
main (void)
{
double data[] = { 1.0 , 1/2.0, 1/3.0, 1/4.0,
1/2.0, 1/3.0, 1/4.0, 1/5.0,
1/3.0, 1/4.0, 1/5.0, 1/6.0,
1/4.0, 1/5.0, 1/6.0, 1/7.0 };
gsl_matrix_view m
= gsl_matrix_view_array (data, 4, 4);
gsl_vector *eval = gsl_vector_alloc (4);
gsl_matrix *evec = gsl_matrix_alloc (4, 4);
gsl_eigen_symmv_workspace * w =
gsl_eigen_symmv_alloc (4);
gsl_eigen_symmv (&m.matrix, eval, evec, w);
gsl_eigen_symmv_free (w);
gsl_eigen_symmv_sort (eval, evec,
GSL_EIGEN_SORT_ABS_ASC);
{
int i;
for (i = 0; i < 4; i++)
{
double eval_i
= gsl_vector_get (eval, i);
gsl_vector_view evec_i
= gsl_matrix_column (evec, i);
printf ("eigenvalue = %g\n", eval_i);
printf ("eigenvector = \n");
gsl_vector_fprintf (stdout,
&evec_i.vector, "%g");
}
}
gsl_vector_free (eval);
gsl_matrix_free (evec);
return 0;
}
Here is the beginning of the output from the program,
$ ./a.out eigenvalue = 9.67023e-05 eigenvector = -0.0291933 0.328712 -0.791411 0.514553 ...
This can be compared with the corresponding output from GNU OCTAVE,
octave> [v,d] = eig(hilb(4)); octave> diag(d) ans = 9.6702e-05 6.7383e-03 1.6914e-01 1.5002e+00 octave> v v = 0.029193 0.179186 -0.582076 0.792608 -0.328712 -0.741918 0.370502 0.451923 0.791411 0.100228 0.509579 0.322416 -0.514553 0.638283 0.514048 0.252161
Note that the eigenvectors can differ by a change of sign, since the sign of an eigenvector is arbitrary.
The following program illustrates the use of the nonsymmetric eigensolver, by computing the eigenvalues and eigenvectors of the Vandermonde matrix V(x;i,j) = x_i^{n - j} with x = (-1,-2,3,4).
#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_eigen.h>
int
main (void)
{
double data[] = { -1.0, 1.0, -1.0, 1.0,
-8.0, 4.0, -2.0, 1.0,
27.0, 9.0, 3.0, 1.0,
64.0, 16.0, 4.0, 1.0 };
gsl_matrix_view m
= gsl_matrix_view_array (data, 4, 4);
gsl_vector_complex *eval = gsl_vector_complex_alloc (4);
gsl_matrix_complex *evec = gsl_matrix_complex_alloc (4, 4);
gsl_eigen_nonsymmv_workspace * w =
gsl_eigen_nonsymmv_alloc (4);
gsl_eigen_nonsymmv (&m.matrix, eval, evec, w);
gsl_eigen_nonsymmv_free (w);
gsl_eigen_nonsymmv_sort (eval, evec,
GSL_EIGEN_SORT_ABS_DESC);
{
int i, j;
for (i = 0; i < 4; i++)
{
gsl_complex eval_i
= gsl_vector_complex_get (eval, i);
gsl_vector_complex_view evec_i
= gsl_matrix_complex_column (evec, i);
printf ("eigenvalue = %g + %gi\n",
GSL_REAL(eval_i), GSL_IMAG(eval_i));
printf ("eigenvector = \n");
for (j = 0; j < 4; ++j)
{
gsl_complex z =
gsl_vector_complex_get(&evec_i.vector, j);
printf("%g + %gi\n", GSL_REAL(z), GSL_IMAG(z));
}
}
}
gsl_vector_complex_free(eval);
gsl_matrix_complex_free(evec);
return 0;
}
Here is the beginning of the output from the program,
$ ./a.out eigenvalue = -6.41391 + 0i eigenvector = -0.0998822 + 0i -0.111251 + 0i 0.292501 + 0i 0.944505 + 0i eigenvalue = 5.54555 + 3.08545i eigenvector = -0.043487 + -0.0076308i 0.0642377 + -0.142127i -0.515253 + 0.0405118i -0.840592 + -0.00148565i ...
This can be compared with the corresponding output from GNU OCTAVE,
octave> [v,d] = eig(vander([-1 -2 3 4])); octave> diag(d) ans = -6.4139 + 0.0000i 5.5456 + 3.0854i 5.5456 - 3.0854i 2.3228 + 0.0000i octave> v v = Columns 1 through 3: -0.09988 + 0.00000i -0.04350 - 0.00755i -0.04350 + 0.00755i -0.11125 + 0.00000i 0.06399 - 0.14224i 0.06399 + 0.14224i 0.29250 + 0.00000i -0.51518 + 0.04142i -0.51518 - 0.04142i 0.94451 + 0.00000i -0.84059 + 0.00000i -0.84059 - 0.00000i Column 4: -0.14493 + 0.00000i 0.35660 + 0.00000i 0.91937 + 0.00000i 0.08118 + 0.00000i
Note that the eigenvectors corresponding to the eigenvalue 5.54555 + 3.08545i differ by the multiplicative constant 0.9999984 + 0.0017674i which is an arbitrary phase factor of magnitude 1.
Next: Eigenvalue and Eigenvector References, Previous: Sorting Eigenvalues and Eigenvectors, Up: Eigensystems [Index]
Next: The Chi-squared Distribution, Previous: The Flat (Uniform) Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the lognormal distribution. The distribution function is,
p(x) dx = {1 \over x \sqrt{2 \pi \sigma^2} } \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx
for x > 0.
This function computes the probability density p(x) at x for a lognormal distribution with parameters zeta and sigma, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the lognormal distribution with parameters zeta and sigma.
Next: The histogram probability distribution struct, Previous: Reading and writing histograms, Up: Histograms [Index]
A histogram made by counting events can be regarded as a measurement of a probability distribution. Allowing for statistical error, the height of each bin represents the probability of an event where the value of x falls in the range of that bin. The probability distribution function has the one-dimensional form p(x)dx where,
p(x) = n_i/ (N w_i)
In this equation n_i is the number of events in the bin which contains x, w_i is the width of the bin and N is the total number of events. The distribution of events within each bin is assumed to be uniform.
gsl-ref-html-2.3/Histogram-Operations.html 0000664 0001750 0001750 00000014566 13055414451 016715 0 ustar edd eddNext: Reading and writing histograms, Previous: Histogram Statistics, Up: Histograms [Index]
This function returns 1 if the all of the individual bin ranges of the two histograms are identical, and 0 otherwise.
This function adds the contents of the bins in histogram h2 to the corresponding bins of histogram h1, i.e. h'_1(i) = h_1(i) + h_2(i). The two histograms must have identical bin ranges.
This function subtracts the contents of the bins in histogram h2 from the corresponding bins of histogram h1, i.e. h'_1(i) = h_1(i) - h_2(i). The two histograms must have identical bin ranges.
This function multiplies the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h'_1(i) = h_1(i) * h_2(i). The two histograms must have identical bin ranges.
This function divides the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h'_1(i) = h_1(i) / h_2(i). The two histograms must have identical bin ranges.
This function multiplies the contents of the bins of histogram h by the constant scale, i.e. h'_1(i) = h_1(i) * scale.
This function shifts the contents of the bins of histogram h by the constant offset, i.e. h'_1(i) = h_1(i) + offset.
Next: Nonlinear Least-Squares Examples, Previous: Nonlinear Least-Squares Covariance Matrix, Up: Nonlinear Least-Squares Fitting [Index]
When developing a code to solve a nonlinear least squares problem, here are a few considerations to keep in mind.
Next: Nonlinear Least-Squares Examples, Previous: Nonlinear Least-Squares Covariance Matrix, Up: Nonlinear Least-Squares Fitting [Index]
Next: Chebyshev Approximation References and Further Reading, Previous: Derivatives and Integrals, Up: Chebyshev Approximations [Index]
The following example program computes Chebyshev approximations to a step function. This is an extremely difficult approximation to make, due to the discontinuity, and was chosen as an example where approximation error is visible. For smooth functions the Chebyshev approximation converges extremely rapidly and errors would not be visible.
#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_chebyshev.h>
double
f (double x, void *p)
{
(void)(p); /* avoid unused parameter warning */
if (x < 0.5)
return 0.25;
else
return 0.75;
}
int
main (void)
{
int i, n = 10000;
gsl_cheb_series *cs = gsl_cheb_alloc (40);
gsl_function F;
F.function = f;
F.params = 0;
gsl_cheb_init (cs, &F, 0.0, 1.0);
for (i = 0; i < n; i++)
{
double x = i / (double)n;
double r10 = gsl_cheb_eval_n (cs, 10, x);
double r40 = gsl_cheb_eval (cs, x);
printf ("%g %g %g %g\n",
x, GSL_FN_EVAL (&F, x), r10, r40);
}
gsl_cheb_free (cs);
return 0;
}
The output from the program gives the original function, 10-th order approximation and 40-th order approximation, all sampled at intervals of 0.001 in x.
gsl-ref-html-2.3/The-Type_002d1-Gumbel-Distribution.html 0000664 0001750 0001750 00000013400 13055414434 020755 0 ustar edd eddNext: The Type-2 Gumbel Distribution, Previous: The Weibull Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the Type-1 Gumbel distribution. The Type-1 Gumbel distribution function is,
p(x) dx = a b \exp(-(b \exp(-ax) + ax)) dx
for -\infty < x < \infty.
This function computes the probability density p(x) at x for a Type-1 Gumbel distribution with parameters a and b, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Type-1 Gumbel distribution with parameters a and b.
Next: Compiling and Linking, Up: Using the library [Index]
The following short program demonstrates the use of the library by computing the value of the Bessel function J_0(x) for x=5,
#include <stdio.h>
#include <gsl/gsl_sf_bessel.h>
int
main (void)
{
double x = 5.0;
double y = gsl_sf_bessel_J0 (x);
printf ("J0(%g) = %.18e\n", x, y);
return 0;
}
The output is shown below, and should be correct to double-precision accuracy,2
J0(5) = -1.775967713143382642e-01
The steps needed to compile this program are described in the following sections.
The last few digits may vary slightly depending on the compiler and platform used—this is normal.
Next: Example programs for Multidimensional Root finding, Previous: Algorithms using Derivatives, Up: Multidimensional Root-Finding [Index]
The algorithms described in this section do not require any derivative information to be supplied by the user. Any derivatives needed are approximated by finite differences. Note that if the finite-differencing step size chosen by these routines is inappropriate, an explicit user-supplied numerical derivative can always be used with the algorithms described in the previous section.
This is a version of the Hybrid algorithm which replaces calls to the
Jacobian function by its finite difference approximation. The finite
difference approximation is computed using gsl_multiroots_fdjac
with a relative step size of GSL_SQRT_DBL_EPSILON. Note that
this step size will not be suitable for all problems.
This is a finite difference version of the Hybrid algorithm without internal scaling.
The discrete Newton algorithm is the simplest method of solving a multidimensional system. It uses the Newton iteration
x -> x - J^{-1} f(x)
where the Jacobian matrix J is approximated by taking finite differences of the function f. The approximation scheme used by this implementation is,
J_{ij} = (f_i(x + \delta_j) - f_i(x)) / \delta_j
where \delta_j is a step of size \sqrt\epsilon |x_j| with \epsilon being the machine precision (\epsilon \approx 2.22 \times 10^-16). The order of convergence of Newton’s algorithm is quadratic, but the finite differences require n^2 function evaluations on each iteration. The algorithm may become unstable if the finite differences are not a good approximation to the true derivatives.
The Broyden algorithm is a version of the discrete Newton algorithm which attempts to avoids the expensive update of the Jacobian matrix on each iteration. The changes to the Jacobian are also approximated, using a rank-1 update,
J^{-1} \to J^{-1} - (J^{-1} df - dx) dx^T J^{-1} / dx^T J^{-1} df
where the vectors dx and df are the changes in x and f. On the first iteration the inverse Jacobian is estimated using finite differences, as in the discrete Newton algorithm.
This approximation gives a fast update but is unreliable if the changes are not small, and the estimate of the inverse Jacobian becomes worse as time passes. The algorithm has a tendency to become unstable unless it starts close to the root. The Jacobian is refreshed if this instability is detected (consult the source for details).
This algorithm is included only for demonstration purposes, and is not recommended for serious use.
Next: Example programs for Multidimensional Root finding, Previous: Algorithms using Derivatives, Up: Multidimensional Root-Finding [Index]
Next: Obtaining GSL, Previous: Routines available in GSL, Up: Introduction [Index]
The subroutines in the GNU Scientific Library are “free software”; this means that everyone is free to use them, and to redistribute them in other free programs. The library is not in the public domain; it is copyrighted and there are conditions on its distribution. These conditions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of the software that they might get from you.
Specifically, we want to make sure that you have the right to share copies of programs that you are given which use the GNU Scientific Library, that you receive their source code or else can get it if you want it, that you can change these programs or use pieces of them in new free programs, and that you know you can do these things.
To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of any code which uses the GNU Scientific Library, you must give the recipients all the rights that you have received. You must make sure that they, too, receive or can get the source code, both to the library and the code which uses it. And you must tell them their rights. This means that the library should not be redistributed in proprietary programs.
Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU Scientific Library. If these programs are modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation.
The precise conditions for the distribution of software related to the GNU Scientific Library are found in the GNU General Public License (see GNU General Public License). Further information about this license is available from the GNU Project webpage Frequently Asked Questions about the GNU GPL,
The Free Software Foundation also operates a license consulting service for commercial users (contact details available from http://www.fsf.org/).
Next: Obtaining GSL, Previous: Routines available in GSL, Up: Introduction [Index]
Previous: Random Number References and Further Reading, Up: Random Number Generation [Index]
Thanks to Makoto Matsumoto, Takuji Nishimura and Yoshiharu Kurita for making the source code to their generators (MT19937, MM&TN; TT800, MM&YK) available under the GNU General Public License. Thanks to Martin Lüscher for providing notes and source code for the RANLXS and RANLXD generators.
gsl-ref-html-2.3/Bessel-Functions.html 0000664 0001750 0001750 00000017171 13055414560 016016 0 ustar edd eddNext: Clausen Functions, Previous: Airy Functions and Derivatives, Up: Special Functions [Index]
The routines described in this section compute the Cylindrical Bessel functions J_n(x), Y_n(x), Modified cylindrical Bessel functions I_n(x), K_n(x), Spherical Bessel functions j_l(x), y_l(x), and Modified Spherical Bessel functions i_l(x), k_l(x). For more information see Abramowitz & Stegun, Chapters 9 and 10. The Bessel functions are defined in the header file gsl_sf_bessel.h.
Next: Mixed-radix FFT routines for real data, Previous: Overview of real data FFTs, Up: Fast Fourier Transforms [Index]
This section describes radix-2 FFT algorithms for real data. They use the Cooley-Tukey algorithm to compute in-place FFTs for lengths which are a power of 2.
The radix-2 FFT functions for real data are declared in the header files gsl_fft_real.h
This function computes an in-place radix-2 FFT of length n and stride stride on the real array data. The output is a half-complex sequence, which is stored in-place. The arrangement of the half-complex terms uses the following scheme: for k < n/2 the real part of the k-th term is stored in location k, and the corresponding imaginary part is stored in location n-k. Terms with k > n/2 can be reconstructed using the symmetry z_k = z^*_{n-k}. The terms for k=0 and k=n/2 are both purely real, and count as a special case. Their real parts are stored in locations 0 and n/2 respectively, while their imaginary parts which are zero are not stored.
The following table shows the correspondence between the output data and the equivalent results obtained by considering the input data as a complex sequence with zero imaginary part (assuming stride=1),
complex[0].real = data[0] complex[0].imag = 0 complex[1].real = data[1] complex[1].imag = data[n-1] ............... ................ complex[k].real = data[k] complex[k].imag = data[n-k] ............... ................ complex[n/2].real = data[n/2] complex[n/2].imag = 0 ............... ................ complex[k'].real = data[k] k' = n - k complex[k'].imag = -data[n-k] ............... ................ complex[n-1].real = data[1] complex[n-1].imag = -data[n-1]
Note that the output data can be converted into the full complex
sequence using the function gsl_fft_halfcomplex_radix2_unpack
described below.
The radix-2 FFT functions for halfcomplex data are declared in the header file gsl_fft_halfcomplex.h.
These functions compute the inverse or backwards in-place radix-2 FFT of
length n and stride stride on the half-complex sequence
data stored according the output scheme used by
gsl_fft_real_radix2. The result is a real array stored in natural
order.
This function converts halfcomplex_coefficient, an array of
half-complex coefficients as returned by gsl_fft_real_radix2_transform, into an ordinary complex array, complex_coefficient. It fills in the
complex array using the symmetry
z_k = z_{n-k}^*
to reconstruct the redundant elements. The algorithm for the conversion
is,
complex_coefficient[0].real
= halfcomplex_coefficient[0];
complex_coefficient[0].imag
= 0.0;
for (i = 1; i < n - i; i++)
{
double hc_real
= halfcomplex_coefficient[i*stride];
double hc_imag
= halfcomplex_coefficient[(n-i)*stride];
complex_coefficient[i*stride].real = hc_real;
complex_coefficient[i*stride].imag = hc_imag;
complex_coefficient[(n - i)*stride].real = hc_real;
complex_coefficient[(n - i)*stride].imag = -hc_imag;
}
if (i == n - i)
{
complex_coefficient[i*stride].real
= halfcomplex_coefficient[(n - 1)*stride];
complex_coefficient[i*stride].imag
= 0.0;
}
Next: Mixed-radix FFT routines for real data, Previous: Overview of real data FFTs, Up: Fast Fourier Transforms [Index]
Next: Permutation properties, Previous: Permutation allocation, Up: Permutations [Index]
The following functions can be used to access and manipulate permutations.
This function returns the value of the i-th element of the
permutation p. If i lies outside the allowed range of 0 to
n-1 then the error handler is invoked and 0 is returned. An inline version of this function is used when HAVE_INLINE is defined.
This function exchanges the i-th and j-th elements of the permutation p.
Next: Accessing combination elements, Previous: The Combination struct, Up: Combinations [Index]
This function allocates memory for a new combination with parameters
n, k. The combination is not initialized and its elements
are undefined. Use the function gsl_combination_calloc if you
want to create a combination which is initialized to the
lexicographically first combination. A null pointer is returned if
insufficient memory is available to create the combination.
This function allocates memory for a new combination with parameters n, k and initializes it to the lexicographically first combination. A null pointer is returned if insufficient memory is available to create the combination.
This function initializes the combination c to the lexicographically first combination, i.e. (0,1,2,…,k-1).
This function initializes the combination c to the lexicographically last combination, i.e. (n-k,n-k+1,…,n-1).
This function frees all the memory used by the combination c.
This function copies the elements of the combination src into the combination dest. The two combinations must have the same size.
Next: Providing the multidimensional system of equations to solve, Previous: Overview of Multidimensional Root Finding, Up: Multidimensional Root-Finding [Index]
The following functions initialize a multidimensional solver, either with or without derivatives. The solver itself depends only on the dimension of the problem and the algorithm and can be reused for different problems.
This function returns a pointer to a newly allocated instance of a solver of type T for a system of n dimensions. For example, the following code creates an instance of a hybrid solver, to solve a 3-dimensional system of equations.
const gsl_multiroot_fsolver_type * T
= gsl_multiroot_fsolver_hybrid;
gsl_multiroot_fsolver * s
= gsl_multiroot_fsolver_alloc (T, 3);
If there is insufficient memory to create the solver then the function
returns a null pointer and the error handler is invoked with an error
code of GSL_ENOMEM.
This function returns a pointer to a newly allocated instance of a derivative solver of type T for a system of n dimensions. For example, the following code creates an instance of a Newton-Raphson solver, for a 2-dimensional system of equations.
const gsl_multiroot_fdfsolver_type * T
= gsl_multiroot_fdfsolver_newton;
gsl_multiroot_fdfsolver * s =
gsl_multiroot_fdfsolver_alloc (T, 2);
If there is insufficient memory to create the solver then the function
returns a null pointer and the error handler is invoked with an error
code of GSL_ENOMEM.
These functions set, or reset, an existing solver s to use the function f or function and derivative fdf, and the initial guess x. Note that the initial position is copied from x, this argument is not modified by subsequent iterations.
These functions free all the memory associated with the solver s.
These functions return a pointer to the name of the solver. For example,
printf ("s is a '%s' solver\n",
gsl_multiroot_fdfsolver_name (s));
would print something like s is a 'newton' solver.
Next: Providing the multidimensional system of equations to solve, Previous: Overview of Multidimensional Root Finding, Up: Multidimensional Root-Finding [Index]
Next: Nonlinear Least-Squares Covariance Matrix, Previous: Nonlinear Least-Squares Testing for Convergence, Up: Nonlinear Least-Squares Fitting [Index]
These routines provide a high level wrapper that combines the iteration and convergence testing for easy use.
These functions iterate the nonlinear least squares solver w for a
maximum of maxiter iterations. After each iteration, the system is
tested for convergence with the error tolerances xtol, gtol and ftol.
Additionally, the user may supply a callback function callback
which is called after each iteration, so that the user may save or print
relevant quantities for each iteration. The parameter callback_params
is passed to the callback function. The parameters callback
and callback_params may be set to NULL to disable this feature.
Upon successful convergence, the function returns GSL_SUCCESS
and sets info to the reason for convergence (see
gsl_multifit_nlinear_test). If the function has not
converged after maxiter iterations, GSL_EMAXITER is
returned. In rare cases, during an iteration the algorithm may
be unable to find a new acceptable step \delta to take. In
this case, GSL_ENOPROG is returned indicating no further
progress can be made. If your problem is having difficulty converging,
see Nonlinear Least-Squares Troubleshooting for further guidance.
Previous: Chebyshev Approximation Examples, Up: Chebyshev Approximations [Index]
The following paper describes the use of Chebyshev series,
Previous: Relative Exponential Functions, Up: Exponential Functions [Index]
This function exponentiates x with an associated absolute error dx.
This function exponentiates a quantity x with an associated absolute
error dx using the gsl_sf_result_e10 type to return a result with
extended range.
This routine computes the product y \exp(x) for the quantities x, y with associated absolute errors dx, dy.
This routine computes the product y \exp(x) for the quantities
x, y with associated absolute errors dx, dy using the
gsl_sf_result_e10 type to return a result with extended range.
Next: Nonlinear Least-Squares Comparison Example, Previous: Nonlinear Least-Squares Exponential Fit Example, Up: Nonlinear Least-Squares Examples [Index]
The following example program minimizes a modified Rosenbrock function, which is characterized by a narrow canyon with steep walls. The starting point is selected high on the canyon wall, so the solver must first find the canyon bottom and then navigate to the minimum. The problem is solved both with and without using geodesic acceleration for comparison. The cost function is given by
Phi(x) = 1/2 (f1^2 + f2^2) f1 = 100 ( x2 - x1^2 ) f2 = 1 - x1
The Jacobian matrix is given by
J = [ -200*x1 100 ; -1 0 ]
In order to use geodesic acceleration, the user must provide the second directional derivative of each residual in the velocity direction, D_v^2 f_i = \sum_{\alpha\beta} v_{\alpha} v_{\beta} \partial_{\alpha} \partial_{\beta} f_i. The velocity vector v is provided by the solver. For this example, these derivatives are given by
fvv = [ -200 v1^2 ; 0 ]
The solution of this minimization problem is given by
x* = [ 1 ; 1 ] Phi(x*) = 0
The program output is shown below.
=== Solving system without acceleration === NITER = 53 NFEV = 56 NJEV = 54 NAEV = 0 initial cost = 2.250225000000e+04 final cost = 6.674986031430e-18 final x = (9.999999974165e-01, 9.999999948328e-01) final cond(J) = 6.000096055094e+02 === Solving system with acceleration === NITER = 15 NFEV = 17 NJEV = 16 NAEV = 16 initial cost = 2.250225000000e+04 final cost = 7.518932873279e-19 final x = (9.999999991329e-01, 9.999999982657e-01) final cond(J) = 6.000097233278e+02
We can see that enabling geodesic acceleration requires less than a third of the number of Jacobian evaluations in order to locate the minimum. The path taken by both methods is shown in the figure below. The contours show the cost function \Phi(x_1,x_2). We see that both methods quickly find the canyon bottom, but the geodesic acceleration method navigates along the bottom to the solution with significantly fewer iterations.
The program is given below.
#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_multifit_nlinear.h>
int
func_f (const gsl_vector * x, void *params, gsl_vector * f)
{
double x1 = gsl_vector_get(x, 0);
double x2 = gsl_vector_get(x, 1);
gsl_vector_set(f, 0, 100.0 * (x2 - x1*x1));
gsl_vector_set(f, 1, 1.0 - x1);
return GSL_SUCCESS;
}
int
func_df (const gsl_vector * x, void *params, gsl_matrix * J)
{
double x1 = gsl_vector_get(x, 0);
gsl_matrix_set(J, 0, 0, -200.0*x1);
gsl_matrix_set(J, 0, 1, 100.0);
gsl_matrix_set(J, 1, 0, -1.0);
gsl_matrix_set(J, 1, 1, 0.0);
return GSL_SUCCESS;
}
int
func_fvv (const gsl_vector * x, const gsl_vector * v,
void *params, gsl_vector * fvv)
{
double v1 = gsl_vector_get(v, 0);
gsl_vector_set(fvv, 0, -200.0 * v1 * v1);
gsl_vector_set(fvv, 1, 0.0);
return GSL_SUCCESS;
}
void
callback(const size_t iter, void *params,
const gsl_multifit_nlinear_workspace *w)
{
gsl_vector * x = gsl_multifit_nlinear_position(w);
/* print out current location */
printf("%f %f\n",
gsl_vector_get(x, 0),
gsl_vector_get(x, 1));
}
void
solve_system(gsl_vector *x0, gsl_multifit_nlinear_fdf *fdf,
gsl_multifit_nlinear_parameters *params)
{
const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
const size_t max_iter = 200;
const double xtol = 1.0e-8;
const double gtol = 1.0e-8;
const double ftol = 1.0e-8;
const size_t n = fdf->n;
const size_t p = fdf->p;
gsl_multifit_nlinear_workspace *work =
gsl_multifit_nlinear_alloc(T, params, n, p);
gsl_vector * f = gsl_multifit_nlinear_residual(work);
gsl_vector * x = gsl_multifit_nlinear_position(work);
int info;
double chisq0, chisq, rcond;
/* initialize solver */
gsl_multifit_nlinear_init(x0, fdf, work);
/* store initial cost */
gsl_blas_ddot(f, f, &chisq0);
/* iterate until convergence */
gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
callback, NULL, &info, work);
/* store final cost */
gsl_blas_ddot(f, f, &chisq);
/* store cond(J(x)) */
gsl_multifit_nlinear_rcond(&rcond, work);
/* print summary */
fprintf(stderr, "NITER = %zu\n", gsl_multifit_nlinear_niter(work));
fprintf(stderr, "NFEV = %zu\n", fdf->nevalf);
fprintf(stderr, "NJEV = %zu\n", fdf->nevaldf);
fprintf(stderr, "NAEV = %zu\n", fdf->nevalfvv);
fprintf(stderr, "initial cost = %.12e\n", chisq0);
fprintf(stderr, "final cost = %.12e\n", chisq);
fprintf(stderr, "final x = (%.12e, %.12e)\n",
gsl_vector_get(x, 0), gsl_vector_get(x, 1));
fprintf(stderr, "final cond(J) = %.12e\n", 1.0 / rcond);
printf("\n\n");
gsl_multifit_nlinear_free(work);
}
int
main (void)
{
const size_t n = 2;
const size_t p = 2;
gsl_vector *f = gsl_vector_alloc(n);
gsl_vector *x = gsl_vector_alloc(p);
gsl_multifit_nlinear_fdf fdf;
gsl_multifit_nlinear_parameters fdf_params =
gsl_multifit_nlinear_default_parameters();
/* print map of Phi(x1, x2) */
{
double x1, x2, chisq;
double *f1 = gsl_vector_ptr(f, 0);
double *f2 = gsl_vector_ptr(f, 1);
for (x1 = -1.2; x1 < 1.3; x1 += 0.1)
{
for (x2 = -0.5; x2 < 2.1; x2 += 0.1)
{
gsl_vector_set(x, 0, x1);
gsl_vector_set(x, 1, x2);
func_f(x, NULL, f);
chisq = (*f1) * (*f1) + (*f2) * (*f2);
printf("%f %f %f\n", x1, x2, chisq);
}
printf("\n");
}
printf("\n\n");
}
/* define function to be minimized */
fdf.f = func_f;
fdf.df = func_df;
fdf.fvv = func_fvv;
fdf.n = n;
fdf.p = p;
fdf.params = NULL;
/* starting point */
gsl_vector_set(x, 0, -0.5);
gsl_vector_set(x, 1, 1.75);
fprintf(stderr, "=== Solving system without acceleration ===\n");
fdf_params.trs = gsl_multifit_nlinear_trs_lm;
solve_system(x, &fdf, &fdf_params);
fprintf(stderr, "=== Solving system with acceleration ===\n");
fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
solve_system(x, &fdf, &fdf_params);
gsl_vector_free(f);
gsl_vector_free(x);
return 0;
}
Next: Nonlinear Least-Squares Comparison Example, Previous: Nonlinear Least-Squares Exponential Fit Example, Up: Nonlinear Least-Squares Examples [Index]
Next: Combination allocation, Up: Combinations [Index]
A combination is defined by a structure containing three components, the
values of n and k, and a pointer to the combination array.
The elements of the combination array are all of type size_t, and
are stored in increasing order. The gsl_combination structure
looks like this,
typedef struct
{
size_t n;
size_t k;
size_t *data;
} gsl_combination;
Next: QAGP adaptive integration with known singular points, Previous: QAG adaptive integration, Up: Numerical Integration [Index]
The presence of an integrable singularity in the integration region causes an adaptive routine to concentrate new subintervals around the singularity. As the subintervals decrease in size the successive approximations to the integral converge in a limiting fashion. This approach to the limit can be accelerated using an extrapolation procedure. The QAGS algorithm combines adaptive bisection with the Wynn epsilon-algorithm to speed up the integration of many types of integrable singularities.
This function applies the Gauss-Kronrod 21-point integration rule adaptively until an estimate of the integral of f over (a,b) is achieved within the desired absolute and relative error limits, epsabs and epsrel. The results are extrapolated using the epsilon-algorithm, which accelerates the convergence of the integral in the presence of discontinuities and integrable singularities. The function returns the final approximation from the extrapolation, result, and an estimate of the absolute error, abserr. The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals is given by limit, which may not exceed the allocated size of the workspace.
Next: Nonlinear Least-Squares TRS Steihaug-Toint Conjugate Gradient, Previous: Nonlinear Least-Squares TRS Double Dogleg, Up: Nonlinear Least-Squares TRS Overview [Index]
The dogleg methods restrict the search for the TRS solution to a 1D curve defined by the Cauchy and Gauss-Newton points. An improvement to this is to search for a solution using the full two dimensional subspace spanned by the Cauchy and Gauss-Newton directions. The dogleg path is of course inside this subspace, and so this method solves the TRS at least as accurately as the dogleg methods. Since this method searches a larger subspace for a solution, it can converge more quickly than dogleg on some problems. Because the subspace is only two dimensional, this method is very efficient and the main computation per iteration is to determine the Gauss-Newton point.
gsl-ref-html-2.3/Running-Statistics-Example-programs.html 0000664 0001750 0001750 00000022774 13055414572 021634 0 ustar edd eddNext: Running Statistics References and Further Reading, Previous: Running Statistics Quantiles, Up: Running Statistics [Index]
Here is a basic example of how to use the statistical functions:
#include <stdio.h>
#include <gsl/gsl_rstat.h>
int
main(void)
{
double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
double mean, variance, largest, smallest, sd,
rms, sd_mean, median, skew, kurtosis;
gsl_rstat_workspace *rstat_p = gsl_rstat_alloc();
size_t i, n;
/* add data to rstat accumulator */
for (i = 0; i < 5; ++i)
gsl_rstat_add(data[i], rstat_p);
mean = gsl_rstat_mean(rstat_p);
variance = gsl_rstat_variance(rstat_p);
largest = gsl_rstat_max(rstat_p);
smallest = gsl_rstat_min(rstat_p);
median = gsl_rstat_median(rstat_p);
sd = gsl_rstat_sd(rstat_p);
sd_mean = gsl_rstat_sd_mean(rstat_p);
skew = gsl_rstat_skew(rstat_p);
rms = gsl_rstat_rms(rstat_p);
kurtosis = gsl_rstat_kurtosis(rstat_p);
n = gsl_rstat_n(rstat_p);
printf ("The dataset is %g, %g, %g, %g, %g\n",
data[0], data[1], data[2], data[3], data[4]);
printf ("The sample mean is %g\n", mean);
printf ("The estimated variance is %g\n", variance);
printf ("The largest value is %g\n", largest);
printf ("The smallest value is %g\n", smallest);
printf( "The median is %g\n", median);
printf( "The standard deviation is %g\n", sd);
printf( "The root mean square is %g\n", rms);
printf( "The standard devation of the mean is %g\n", sd_mean);
printf( "The skew is %g\n", skew);
printf( "The kurtosis %g\n", kurtosis);
printf( "There are %zu items in the accumulator\n", n);
gsl_rstat_reset(rstat_p);
n = gsl_rstat_n(rstat_p);
printf( "There are %zu items in the accumulator\n", n);
gsl_rstat_free(rstat_p);
return 0;
}
The program should produce the following output,
The dataset is 17.2, 18.1, 16.5, 18.3, 12.6 The sample mean is 16.54 The estimated variance is 5.373 The largest value is 18.3 The smallest value is 12.6 The median is 16.5 The standard deviation is 2.31797 The root mean square is 16.6694 The standard devation of the mean is 1.03663 The skew is -0.829058 The kurtosis -1.2217 There are 5 items in the accumulator There are 0 items in the accumulator
This next program estimates the lower quartile, median and upper quartile from 10,000 samples of a random Rayleigh distribution, using the P^2 algorithm of Jain and Chlamtec. For comparison, the exact values are also computed from the sorted dataset.
#include <stdio.h>
#include <stdlib.h>
#include <gsl/gsl_rstat.h>
#include <gsl/gsl_statistics.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_sort.h>
int
main(void)
{
const size_t N = 10000;
double *data = malloc(N * sizeof(double));
gsl_rstat_quantile_workspace *work_25 = gsl_rstat_quantile_alloc(0.25);
gsl_rstat_quantile_workspace *work_50 = gsl_rstat_quantile_alloc(0.5);
gsl_rstat_quantile_workspace *work_75 = gsl_rstat_quantile_alloc(0.75);
gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
double exact_p25, exact_p50, exact_p75;
double val_p25, val_p50, val_p75;
size_t i;
/* add data to quantile accumulators; also store data for exact
* comparisons */
for (i = 0; i < N; ++i)
{
data[i] = gsl_ran_rayleigh(r, 1.0);
gsl_rstat_quantile_add(data[i], work_25);
gsl_rstat_quantile_add(data[i], work_50);
gsl_rstat_quantile_add(data[i], work_75);
}
/* exact values */
gsl_sort(data, 1, N);
exact_p25 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.25);
exact_p50 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.5);
exact_p75 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.75);
/* estimated values */
val_p25 = gsl_rstat_quantile_get(work_25);
val_p50 = gsl_rstat_quantile_get(work_50);
val_p75 = gsl_rstat_quantile_get(work_75);
printf ("The dataset is %g, %g, %g, %g, %g, ...\n",
data[0], data[1], data[2], data[3], data[4]);
printf ("0.25 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
exact_p25, val_p25, (val_p25 - exact_p25) / exact_p25);
printf ("0.50 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
exact_p50, val_p50, (val_p50 - exact_p50) / exact_p50);
printf ("0.75 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
exact_p75, val_p75, (val_p75 - exact_p75) / exact_p75);
gsl_rstat_quantile_free(work_25);
gsl_rstat_quantile_free(work_50);
gsl_rstat_quantile_free(work_75);
gsl_rng_free(r);
free(data);
return 0;
}
The program should produce the following output,
The dataset is 0.00645272, 0.0074002, 0.0120706, 0.0207256, 0.0227282, ... 0.25 quartile: exact = 0.75766, estimated = 0.75580, error = -2.450209e-03 0.50 quartile: exact = 1.17508, estimated = 1.17438, error = -5.995912e-04 0.75 quartile: exact = 1.65347, estimated = 1.65696, error = 2.110571e-03
Next: Running Statistics References and Further Reading, Previous: Running Statistics Quantiles, Up: Running Statistics [Index]
Next: Zeros of Regular Bessel Functions, Previous: Regular Modified Bessel Functions - Fractional Order, Up: Bessel Functions [Index]
These routines compute the irregular modified Bessel function of fractional order \nu, K_\nu(x) for x>0, \nu>0.
These routines compute the logarithm of the irregular modified Bessel function of fractional order \nu, \ln(K_\nu(x)) for x>0, \nu>0.
These routines compute the scaled irregular modified Bessel function of fractional order \nu, \exp(+|x|) K_\nu(x) for x>0, \nu>0.
Next: Simulated Annealing, Previous: N-tuples, Up: Top [Index]
This chapter describes routines for multidimensional Monte Carlo integration. These include the traditional Monte Carlo method and adaptive algorithms such as VEGAS and MISER which use importance sampling and stratified sampling techniques. Each algorithm computes an estimate of a multidimensional definite integral of the form,
I = \int_xl^xu dx \int_yl^yu dy ... f(x, y, ...)
over a hypercubic region ((x_l,x_u), (y_l,y_u), ...) using a fixed number of function calls. The routines also provide a statistical estimate of the error on the result. This error estimate should be taken as a guide rather than as a strict error bound—random sampling of the region may not uncover all the important features of the function, resulting in an underestimate of the error.
The functions are defined in separate header files for each routine, gsl_monte_plain.h, gsl_monte_miser.h and gsl_monte_vegas.h.
| • Monte Carlo Interface: | ||
| • PLAIN Monte Carlo: | ||
| • MISER: | ||
| • VEGAS: | ||
| • Monte Carlo Examples: | ||
| • Monte Carlo Integration References and Further Reading: |
Next: Ordinary Differential Equations, Previous: Monte Carlo Integration, Up: Top [Index]
Stochastic search techniques are used when the structure of a space is not well understood or is not smooth, so that techniques like Newton’s method (which requires calculating Jacobian derivative matrices) cannot be used. In particular, these techniques are frequently used to solve combinatorial optimization problems, such as the traveling salesman problem.
The goal is to find a point in the space at which a real valued energy function (or cost function) is minimized. Simulated annealing is a minimization technique which has given good results in avoiding local minima; it is based on the idea of taking a random walk through the space at successively lower temperatures, where the probability of taking a step is given by a Boltzmann distribution.
The functions described in this chapter are declared in the header file gsl_siman.h.
| • Simulated Annealing algorithm: | ||
| • Simulated Annealing functions: | ||
| • Examples with Simulated Annealing: | ||
| • Simulated Annealing References and Further Reading: |
Next: Random Number Distribution References and Further Reading, Previous: Shuffling and Sampling, Up: Random Number Distributions [Index]
The following program demonstrates the use of a random number generator to produce variates from a distribution. It prints 10 samples from the Poisson distribution with a mean of 3.
#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
int i, n = 10;
double mu = 3.0;
/* create a generator chosen by the
environment variable GSL_RNG_TYPE */
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
/* print n random variates chosen from
the poisson distribution with mean
parameter mu */
for (i = 0; i < n; i++)
{
unsigned int k = gsl_ran_poisson (r, mu);
printf (" %u", k);
}
printf ("\n");
gsl_rng_free (r);
return 0;
}
If the library and header files are installed under /usr/local (the default location) then the program can be compiled with these options,
$ gcc -Wall demo.c -lgsl -lgslcblas -lm
Here is the output of the program,
$ ./a.out
2 5 5 2 1 0 3 4 1 1
The variates depend on the seed used by the generator. The seed for the
default generator type gsl_rng_default can be changed with the
GSL_RNG_SEED environment variable to produce a different stream
of variates,
$ GSL_RNG_SEED=123 ./a.out
4 5 6 3 3 1 4 2 5 5
The following program generates a random walk in two dimensions.
#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
int
main (void)
{
int i;
double x = 0, y = 0, dx, dy;
const gsl_rng_type * T;
gsl_rng * r;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
printf ("%g %g\n", x, y);
for (i = 0; i < 10; i++)
{
gsl_ran_dir_2d (r, &dx, &dy);
x += dx; y += dy;
printf ("%g %g\n", x, y);
}
gsl_rng_free (r);
return 0;
}
Here is some output from the program, four 10-step random walks from the origin,
The following program computes the upper and lower cumulative distribution functions for the standard normal distribution at x=2.
#include <stdio.h>
#include <gsl/gsl_cdf.h>
int
main (void)
{
double P, Q;
double x = 2.0;
P = gsl_cdf_ugaussian_P (x);
printf ("prob(x < %f) = %f\n", x, P);
Q = gsl_cdf_ugaussian_Q (x);
printf ("prob(x > %f) = %f\n", x, Q);
x = gsl_cdf_ugaussian_Pinv (P);
printf ("Pinv(%f) = %f\n", P, x);
x = gsl_cdf_ugaussian_Qinv (Q);
printf ("Qinv(%f) = %f\n", Q, x);
return 0;
}
Here is the output of the program,
prob(x < 2.000000) = 0.977250 prob(x > 2.000000) = 0.022750 Pinv(0.977250) = 2.000000 Qinv(0.022750) = 2.000000
Next: Random Number Distribution References and Further Reading, Previous: Shuffling and Sampling, Up: Random Number Distributions [Index]
Next: The Poisson Distribution, Previous: The Dirichlet Distribution, Up: Random Number Distributions [Index]
Given K discrete events with different probabilities P[k], produce a random value k consistent with its probability.
The obvious way to do this is to preprocess the probability list by generating a cumulative probability array with K+1 elements:
C[0] = 0 C[k+1] = C[k]+P[k].
Note that this construction produces C[K]=1. Now choose a uniform deviate u between 0 and 1, and find the value of k such that C[k] <= u < C[k+1]. Although this in principle requires of order \log K steps per random number generation, they are fast steps, and if you use something like \lfloor uK \rfloor as a starting point, you can often do pretty well.
But faster methods have been devised. Again, the idea is to preprocess the probability list, and save the result in some form of lookup table; then the individual calls for a random discrete event can go rapidly. An approach invented by G. Marsaglia (Generating discrete random variables in a computer, Comm ACM 6, 37–38 (1963)) is very clever, and readers interested in examples of good algorithm design are directed to this short and well-written paper. Unfortunately, for large K, Marsaglia’s lookup table can be quite large.
A much better approach is due to Alastair J. Walker (An efficient method for generating discrete random variables with general distributions, ACM Trans on Mathematical Software 3, 253–256 (1977); see also Knuth, v2, 3rd ed, p120–121,139). This requires two lookup tables, one floating point and one integer, but both only of size K. After preprocessing, the random numbers are generated in O(1) time, even for large K. The preprocessing suggested by Walker requires O(K^2) effort, but that is not actually necessary, and the implementation provided here only takes O(K) effort. In general, more preprocessing leads to faster generation of the individual random numbers, but a diminishing return is reached pretty early. Knuth points out that the optimal preprocessing is combinatorially difficult for large K.
This method can be used to speed up some of the discrete random number generators below, such as the binomial distribution. To use it for something like the Poisson Distribution, a modification would have to be made, since it only takes a finite set of K outcomes.
This function returns a pointer to a structure that contains the lookup
table for the discrete random number generator. The array P[] contains
the probabilities of the discrete events; these array elements must all be
positive, but they needn’t add up to one (so you can think of them more
generally as “weights”)—the preprocessor will normalize appropriately.
This return value is used
as an argument for the gsl_ran_discrete function below.
After the preprocessor, above, has been called, you use this function to get the discrete random numbers.
Returns the probability P[k] of observing the variable k. Since P[k] is not stored as part of the lookup table, it must be recomputed; this computation takes O(K), so if K is large and you care about the original array P[k] used to create the lookup table, then you should just keep this original array P[k] around.
De-allocates the lookup table pointed to by g.
Next: The Poisson Distribution, Previous: The Dirichlet Distribution, Up: Random Number Distributions [Index]
Next: Numerical Differentiation, Previous: Ordinary Differential Equations, Up: Top [Index]
This chapter describes functions for performing interpolation. The library provides a variety of interpolation methods, including Cubic, Akima, and Steffen splines. The interpolation types are interchangeable, allowing different methods to be used without recompiling. Interpolations can be defined for both normal and periodic boundary conditions. Additional functions are available for computing derivatives and integrals of interpolating functions. Routines are provided for interpolating both one and two dimensional datasets.
These interpolation methods produce curves that pass through each datapoint. To interpolate noisy data with a smoothing curve see Basis Splines.
The functions described in this section are declared in the header files gsl_interp.h and gsl_spline.h.
Next: Numerical Differentiation, Previous: Ordinary Differential Equations, Up: Top [Index]
Next: Hessenberg-Triangular Decomposition of Real Matrices, Previous: Tridiagonal Decomposition of Hermitian Matrices, Up: Linear Algebra [Index]
A general real matrix A can be decomposed by orthogonal similarity transformations into the form
A = U H U^T
where U is orthogonal and H is an upper Hessenberg matrix, meaning that it has zeros below the first subdiagonal. The Hessenberg reduction is the first step in the Schur decomposition for the nonsymmetric eigenvalue problem, but has applications in other areas as well.
This function computes the Hessenberg decomposition of the matrix A by applying the similarity transformation H = U^T A U. On output, H is stored in the upper portion of A. The information required to construct the matrix U is stored in the lower triangular portion of A. U is a product of N - 2 Householder matrices. The Householder vectors are stored in the lower portion of A (below the subdiagonal) and the Householder coefficients are stored in the vector tau. tau must be of length N.
This function constructs the orthogonal matrix U from the
information stored in the Hessenberg matrix H along with the
vector tau. H and tau are outputs from
gsl_linalg_hessenberg_decomp.
This function is similar to gsl_linalg_hessenberg_unpack, except
it accumulates the matrix U into V, so that V' = VU.
The matrix V must be initialized prior to calling this function.
Setting V to the identity matrix provides the same result as
gsl_linalg_hessenberg_unpack. If H is order N, then
V must have N columns but may have any number of rows.
This function sets the lower triangular portion of H, below
the subdiagonal, to zero. It is useful for clearing out the
Householder vectors after calling gsl_linalg_hessenberg_decomp.
Next: Hessenberg-Triangular Decomposition of Real Matrices, Previous: Tridiagonal Decomposition of Hermitian Matrices, Up: Linear Algebra [Index]
Next: Combination Examples, Previous: Combination functions, Up: Combinations [Index]
The library provides functions for reading and writing combinations to a file as binary data or formatted text.
This function writes the elements of the combination c to the
stream stream in binary format. The function returns
GSL_EFAILED if there was a problem writing to the file. Since the
data is written in the native binary format it may not be portable
between different architectures.
This function reads elements from the open stream stream into the
combination c in binary format. The combination c must be
preallocated with correct values of n and k since the
function uses the size of c to determine how many bytes to read.
The function returns GSL_EFAILED if there was a problem reading
from the file. The data is assumed to have been written in the native
binary format on the same architecture.
This function writes the elements of the combination c
line-by-line to the stream stream using the format specifier
format, which should be suitable for a type of size_t.
In ISO C99 the type modifier z represents size_t, so
"%zu\n" is a suitable format.10 The function returns
GSL_EFAILED if there was a problem writing to the file.
This function reads formatted data from the stream stream into the
combination c. The combination c must be preallocated with
correct values of n and k since the function uses the size of c to
determine how many numbers to read. The function returns
GSL_EFAILED if there was a problem reading from the file.
In versions of the
GNU C library prior to the ISO C99 standard,
the type modifier Z was used instead.
Next: Combination Examples, Previous: Combination functions, Up: Combinations [Index]
Next: 1D Index Look-up and Acceleration, Previous: 1D Interpolation Functions, Up: Interpolation [Index]
The interpolation library provides the following interpolation types:
Linear interpolation. This interpolation method does not require any additional memory.
Polynomial interpolation. This method should only be used for interpolating small numbers of points because polynomial interpolation introduces large oscillations, even for well-behaved datasets. The number of terms in the interpolating polynomial is equal to the number of points.
Cubic spline with natural boundary conditions. The resulting curve is piecewise cubic on each interval, with matching first and second derivatives at the supplied data-points. The second derivative is chosen to be zero at the first point and last point.
Cubic spline with periodic boundary conditions. The resulting curve is piecewise cubic on each interval, with matching first and second derivatives at the supplied data-points. The derivatives at the first and last points are also matched. Note that the last point in the data must have the same y-value as the first point, otherwise the resulting periodic interpolation will have a discontinuity at the boundary.
Non-rounded Akima spline with natural boundary conditions. This method uses the non-rounded corner algorithm of Wodicka.
Non-rounded Akima spline with periodic boundary conditions. This method uses the non-rounded corner algorithm of Wodicka.
Steffen’s method guarantees the monotonicity of the interpolating function between the given data points. Therefore, minima and maxima can only occur exactly at the data points, and there can never be spurious oscillations between data points. The interpolated function is piecewise cubic in each interval. The resulting curve and its first derivative are guaranteed to be continuous, but the second derivative may be discontinuous.
The following related functions are available:
This function returns the name of the interpolation type used by interp. For example,
printf ("interp uses '%s' interpolation.\n",
gsl_interp_name (interp));
would print something like,
interp uses 'cspline' interpolation.
These functions return the minimum number of points required by the interpolation object interp or interpolation type T. For example, Akima spline interpolation requires a minimum of 5 points.
Next: 1D Index Look-up and Acceleration, Previous: 1D Interpolation Functions, Up: Interpolation [Index]
Next: Creating row and column views, Previous: Reading and writing matrices, Up: Matrices [Index]
A matrix view is a temporary object, stored on the stack, which can be
used to operate on a subset of matrix elements. Matrix views can be
defined for both constant and non-constant matrices using separate types
that preserve constness. A matrix view has the type
gsl_matrix_view and a constant matrix view has the type
gsl_matrix_const_view. In both cases the elements of the view
can by accessed using the matrix component of the view object. A
pointer gsl_matrix * or const gsl_matrix * can be obtained
by taking the address of the matrix component with the &
operator. In addition to matrix views it is also possible to create
vector views of a matrix, such as row or column views.
These functions return a matrix view of a submatrix of the matrix m. The upper-left element of the submatrix is the element (k1,k2) of the original matrix. The submatrix has n1 rows and n2 columns. The physical number of columns in memory given by tda is unchanged. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j]
where the index i runs from 0 to n1-1 and the index j
runs from 0 to n2-1.
The data pointer of the returned matrix struct is set to null if
the combined parameters (i,j,n1,n2,tda)
overrun the ends of the original matrix.
The new matrix view is only a view of the block underlying the existing matrix, m. The block containing the elements of m is not owned by the new matrix view. When the view goes out of scope the original matrix m and its block will continue to exist. The original memory can only be deallocated by freeing the original matrix. Of course, the original matrix should not be deallocated while the view is still in use.
The function gsl_matrix_const_submatrix is equivalent to
gsl_matrix_submatrix but can be used for matrices which are
declared const.
These functions return a matrix view of the array base. The matrix has n1 rows and n2 columns. The physical number of columns in memory is also given by n2. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = base[i*n2 + j]
where the index i runs from 0 to n1-1 and the index j
runs from 0 to n2-1.
The new matrix is only a view of the array base. When the view goes out of scope the original array base will continue to exist. The original memory can only be deallocated by freeing the original array. Of course, the original array should not be deallocated while the view is still in use.
The function gsl_matrix_const_view_array is equivalent to
gsl_matrix_view_array but can be used for matrices which are
declared const.
These functions return a matrix view of the array base with a physical number of columns tda which may differ from the corresponding dimension of the matrix. The matrix has n1 rows and n2 columns, and the physical number of columns in memory is given by tda. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = base[i*tda + j]
where the index i runs from 0 to n1-1 and the index j
runs from 0 to n2-1.
The new matrix is only a view of the array base. When the view goes out of scope the original array base will continue to exist. The original memory can only be deallocated by freeing the original array. Of course, the original array should not be deallocated while the view is still in use.
The function gsl_matrix_const_view_array_with_tda is equivalent
to gsl_matrix_view_array_with_tda but can be used for matrices
which are declared const.
These functions return a matrix view of the vector v. The matrix has n1 rows and n2 columns. The vector must have unit stride. The physical number of columns in memory is also given by n2. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = v->data[i*n2 + j]
where the index i runs from 0 to n1-1 and the index j
runs from 0 to n2-1.
The new matrix is only a view of the vector v. When the view goes out of scope the original vector v will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use.
The function gsl_matrix_const_view_vector is equivalent to
gsl_matrix_view_vector but can be used for matrices which are
declared const.
These functions return a matrix view of the vector v with a physical number of columns tda which may differ from the corresponding matrix dimension. The vector must have unit stride. The matrix has n1 rows and n2 columns, and the physical number of columns in memory is given by tda. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = v->data[i*tda + j]
where the index i runs from 0 to n1-1 and the index j
runs from 0 to n2-1.
The new matrix is only a view of the vector v. When the view goes out of scope the original vector v will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use.
The function gsl_matrix_const_view_vector_with_tda is equivalent
to gsl_matrix_view_vector_with_tda but can be used for matrices
which are declared const.
Next: Creating row and column views, Previous: Reading and writing matrices, Up: Matrices [Index]
Next: Closing an ntuple file, Previous: Writing ntuples, Up: N-tuples [Index]
This function reads the current row of the ntuple file for ntuple and stores the values in ntuple->data.
Next: Fitting regularized linear regression example 1, Previous: Fitting linear regression example, Up: Fitting Examples [Index]
The following program performs a quadratic fit y = c_0 + c_1 x + c_2
x^2 to a weighted dataset using the generalised linear fitting function
gsl_multifit_wlinear. The model matrix X for a quadratic
fit is given by,
X = [ 1 , x_0 , x_0^2 ;
1 , x_1 , x_1^2 ;
1 , x_2 , x_2^2 ;
... , ... , ... ]
where the column of ones corresponds to the constant term c_0. The two remaining columns corresponds to the terms c_1 x and c_2 x^2.
The program reads n lines of data in the format (x, y, err) where err is the error (standard deviation) in the value y.
#include <stdio.h>
#include <gsl/gsl_multifit.h>
int
main (int argc, char **argv)
{
int i, n;
double xi, yi, ei, chisq;
gsl_matrix *X, *cov;
gsl_vector *y, *w, *c;
if (argc != 2)
{
fprintf (stderr,"usage: fit n < data\n");
exit (-1);
}
n = atoi (argv[1]);
X = gsl_matrix_alloc (n, 3);
y = gsl_vector_alloc (n);
w = gsl_vector_alloc (n);
c = gsl_vector_alloc (3);
cov = gsl_matrix_alloc (3, 3);
for (i = 0; i < n; i++)
{
int count = fscanf (stdin, "%lg %lg %lg",
&xi, &yi, &ei);
if (count != 3)
{
fprintf (stderr, "error reading file\n");
exit (-1);
}
printf ("%g %g +/- %g\n", xi, yi, ei);
gsl_matrix_set (X, i, 0, 1.0);
gsl_matrix_set (X, i, 1, xi);
gsl_matrix_set (X, i, 2, xi*xi);
gsl_vector_set (y, i, yi);
gsl_vector_set (w, i, 1.0/(ei*ei));
}
{
gsl_multifit_linear_workspace * work
= gsl_multifit_linear_alloc (n, 3);
gsl_multifit_wlinear (X, w, y, c, cov,
&chisq, work);
gsl_multifit_linear_free (work);
}
#define C(i) (gsl_vector_get(c,(i)))
#define COV(i,j) (gsl_matrix_get(cov,(i),(j)))
{
printf ("# best fit: Y = %g + %g X + %g X^2\n",
C(0), C(1), C(2));
printf ("# covariance matrix:\n");
printf ("[ %+.5e, %+.5e, %+.5e \n",
COV(0,0), COV(0,1), COV(0,2));
printf (" %+.5e, %+.5e, %+.5e \n",
COV(1,0), COV(1,1), COV(1,2));
printf (" %+.5e, %+.5e, %+.5e ]\n",
COV(2,0), COV(2,1), COV(2,2));
printf ("# chisq = %g\n", chisq);
}
gsl_matrix_free (X);
gsl_vector_free (y);
gsl_vector_free (w);
gsl_vector_free (c);
gsl_matrix_free (cov);
return 0;
}
A suitable set of data for fitting can be generated using the following program. It outputs a set of points with gaussian errors from the curve y = e^x in the region 0 < x < 2.
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_randist.h>
int
main (void)
{
double x;
const gsl_rng_type * T;
gsl_rng * r;
gsl_rng_env_setup ();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
for (x = 0.1; x < 2; x+= 0.1)
{
double y0 = exp (x);
double sigma = 0.1 * y0;
double dy = gsl_ran_gaussian (r, sigma);
printf ("%g %g %g\n", x, y0 + dy, sigma);
}
gsl_rng_free(r);
return 0;
}
The data can be prepared by running the resulting executable program,
$ GSL_RNG_TYPE=mt19937_1999 ./generate > exp.dat $ more exp.dat 0.1 0.97935 0.110517 0.2 1.3359 0.12214 0.3 1.52573 0.134986 0.4 1.60318 0.149182 0.5 1.81731 0.164872 0.6 1.92475 0.182212 ....
To fit the data use the previous program, with the number of data points given as the first argument. In this case there are 19 data points.
$ ./fit 19 < exp.dat 0.1 0.97935 +/- 0.110517 0.2 1.3359 +/- 0.12214 ... # best fit: Y = 1.02318 + 0.956201 X + 0.876796 X^2 # covariance matrix: [ +1.25612e-02, -3.64387e-02, +1.94389e-02 -3.64387e-02, +1.42339e-01, -8.48761e-02 +1.94389e-02, -8.48761e-02, +5.60243e-02 ] # chisq = 23.0987
The parameters of the quadratic fit match the coefficients of the expansion of e^x, taking into account the errors on the parameters and the O(x^3) difference between the exponential and quadratic functions for the larger values of x. The errors on the parameters are given by the square-root of the corresponding diagonal elements of the covariance matrix. The chi-squared per degree of freedom is 1.4, indicating a reasonable fit to the data.
Next: Fitting regularized linear regression example 1, Previous: Fitting linear regression example, Up: Fitting Examples [Index]
Next: One dimensional Minimization, Previous: Discrete Hankel Transforms, Up: Top [Index]
This chapter describes routines for finding roots of arbitrary one-dimensional functions. The library provides low level components for a variety of iterative solvers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the iteration. Each class of methods uses the same framework, so that you can switch between solvers at runtime without needing to recompile your program. Each instance of a solver keeps track of its own state, allowing the solvers to be used in multi-threaded programs.
The header file gsl_roots.h contains prototypes for the root finding functions and related declarations.
Previous: Permutation Examples, Up: Permutations [Index]
The subject of permutations is covered extensively in Knuth’s Sorting and Searching,
For the definition of the canonical form see,
Next: Initializing the Minimizer, Previous: Minimization Overview, Up: One dimensional Minimization [Index]
Note that minimization functions can only search for one minimum at a time. When there are several minima in the search area, the first minimum to be found will be returned; however it is difficult to predict which of the minima this will be. In most cases, no error will be reported if you try to find a minimum in an area where there is more than one.
With all minimization algorithms it can be difficult to determine the location of the minimum to full numerical precision. The behavior of the function in the region of the minimum x^* can be approximated by a Taylor expansion,
y = f(x^*) + (1/2) f''(x^*) (x - x^*)^2
and the second term of this expansion can be lost when added to the first term at finite precision. This magnifies the error in locating x^*, making it proportional to \sqrt \epsilon (where \epsilon is the relative accuracy of the floating point numbers). For functions with higher order minima, such as x^4, the magnification of the error is correspondingly worse. The best that can be achieved is to converge to the limit of numerical accuracy in the function values, rather than the location of the minimum itself.
gsl-ref-html-2.3/Discrete-Hankel-Transforms.html 0000664 0001750 0001750 00000011210 13055414423 017713 0 ustar edd eddNext: One dimensional Root-Finding, Previous: Wavelet Transforms, Up: Top [Index]
This chapter describes functions for performing Discrete Hankel Transforms (DHTs). The functions are declared in the header file gsl_dht.h.
| • Discrete Hankel Transform Definition: | ||
| • Discrete Hankel Transform Functions: | ||
| • Discrete Hankel Transform References: |
Next: Mathieu Function Characteristic Values, Up: Mathieu Functions [Index]
The Mathieu functions can be computed for a single order or for multiple orders, using array-based routines. The array-based routines require a preallocated workspace.
This function returns a workspace for the array versions of the Mathieu routines. The arguments n and qmax specify the maximum order and q-value of Mathieu functions which can be computed with this workspace.
This function frees the workspace work.
Next: IEEE floating-point arithmetic, Previous: Sparse Linear Algebra, Up: Top [Index]
This chapter describes macros for the values of physical constants, such as the speed of light, c, and gravitational constant, G. The values are available in different unit systems, including the standard MKSA system (meters, kilograms, seconds, amperes) and the CGSM system (centimeters, grams, seconds, gauss), which is commonly used in Astronomy.
The definitions of constants in the MKSA system are available in the file gsl_const_mksa.h. The constants in the CGSM system are defined in gsl_const_cgsm.h. Dimensionless constants, such as the fine structure constant, which are pure numbers are defined in gsl_const_num.h.
The full list of constants is described briefly below. Consult the header files themselves for the values of the constants used in the library.
Next: IEEE floating-point arithmetic, Previous: Sparse Linear Algebra, Up: Top [Index]
Next: General Polynomial Equations, Previous: Quadratic Equations, Up: Polynomials [Index]
This function finds the real roots of the cubic equation,
x^3 + a x^2 + b x + c = 0
with a leading coefficient of unity. The number of real roots (either one or three) is returned, and their locations are stored in x0, x1 and x2. If one real root is found then only x0 is modified. When three real roots are found they are stored in x0, x1 and x2 in ascending order. The case of coincident roots is not considered special. For example, the equation (x-1)^3=0 will have three roots with exactly equal values. As in the quadratic case, finite precision may cause equal or closely-spaced real roots to move off the real axis into the complex plane, leading to a discrete change in the number of real roots.
This function finds the complex roots of the cubic equation,
z^3 + a z^2 + b z + c = 0
The number of complex roots is returned (always three) and the locations of the roots are stored in z0, z1 and z2. The roots are returned in ascending order, sorted first by their real components and then by their imaginary components.
Next: Random Number Generation, Previous: Fast Fourier Transforms, Up: Top [Index]
This chapter describes routines for performing numerical integration (quadrature) of a function in one dimension. There are routines for adaptive and non-adaptive integration of general functions, with specialised routines for specific cases. These include integration over infinite and semi-infinite ranges, singular integrals, including logarithmic singularities, computation of Cauchy principal values and oscillatory integrals. The library reimplements the algorithms used in QUADPACK, a numerical integration package written by Piessens, de Doncker-Kapenga, Ueberhuber and Kahaner. Fortran code for QUADPACK is available on Netlib. Also included are non-adaptive, fixed-order Gauss-Legendre integration routines with high precision coefficients by Pavel Holoborodko.
The functions described in this chapter are declared in the header file gsl_integration.h.
Next: Random Number Generation, Previous: Fast Fourier Transforms, Up: Top [Index]
The problem of multidimensional nonlinear least-squares fitting requires the minimization of the squared residuals of n functions, f_i, in p parameters, x_i,
\Phi(x) = (1/2) || f(x) ||^2
= (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2
In trust region methods, the objective (or cost) function \Phi(x) is approximated by a model function m_k(\delta) in the vicinity of some point x_k. The model function is often simply a second order Taylor series expansion around the point x_k, ie:
\Phi(x_k + \delta) ~=~ m_k(\delta) = \Phi(x_k) + g_k^T \delta + 1/2 \delta^T B_k \delta
where g_k = \nabla \Phi(x_k) = J^T f is the gradient vector at the point x_k, B_k = \nabla^2 \Phi(x_k) is the Hessian matrix at x_k, or some approximation to it, and J is the n-by-p Jacobian matrix J_{ij} = d f_i / d x_j. In order to find the next step \delta, we minimize the model function m_k(\delta), but search for solutions only within a region where we trust that m_k(\delta) is a good approximation to the objective function \Phi(x_k + \delta). In other words, we seek a solution of the trust region subproblem (TRS)
\min_(\delta \in R^p) m_k(\delta), s.t. || D_k \delta || <= \Delta_k
where \Delta_k > 0 is the trust region radius and D_k is a scaling matrix. If D_k = I, then the trust region is a ball of radius \Delta_k centered at x_k. In some applications, the parameter vector x may have widely different scales. For example, one parameter might be a temperature on the order of 10^3 K, while another might be a length on the order of 10^{-6} m. In such cases, a spherical trust region may not be the best choice, since if \Phi changes rapidly along directions with one scale, and more slowly along directions with a different scale, the model function m_k may be a poor approximation to \Phi along the rapidly changing directions. In such problems, it may be best to use an elliptical trust region, by setting D_k to a diagonal matrix whose entries are designed so that the scaled step D_k \delta has entries of approximately the same order of magnitude.
The trust region subproblem above normally amounts to solving a linear least squares system (or multiple systems) for the step \delta. Once \delta is computed, it is checked whether or not it reduces the objective function \Phi(x). A useful statistic for this is to look at the ratio
\rho_k = ( \Phi(x_k) - \Phi(x_k + \delta_k) / ( m_k(0) - m_k(\delta_k) )
where the numerator is the actual reduction of the objective function due to the step \delta_k, and the denominator is the predicted reduction due to the model m_k. If \rho_k is negative, it means that the step \delta_k increased the objective function and so it is rejected. If \rho_k is positive, then we have found a step which reduced the objective function and it is accepted. Furthermore, if \rho_k is close to 1, then this indicates that the model function is a good approximation to the objective function in the trust region, and so on the next iteration the trust region is enlarged in order to take more ambitious steps. When a step is rejected, the trust region is made smaller and the TRS is solved again. An outline for the general trust region method used by GSL can now be given.
Trust Region Algorithm
GSL offers the user a number of different algorithms for solving the trust region subproblem in 2(b), as well as different choices of scaling matrices D_k and different methods of updating the trust region radius \Delta_k. Therefore, while reasonable default methods are provided, the user has a lot of control to fine-tune the various steps of the algorithm for their specific problem.
Next: Searching 2D histogram ranges, Previous: Copying 2D Histograms, Up: Histograms [Index]
You can access the bins of a two-dimensional histogram either by specifying a pair of (x,y) coordinates or by using the bin indices (i,j) directly. The functions for accessing the histogram through (x,y) coordinates use binary searches in the x and y directions to identify the bin which covers the appropriate range.
This function updates the histogram h by adding one (1.0) to the bin whose x and y ranges contain the coordinates (x,y).
If the point (x,y) lies inside the valid ranges of the
histogram then the function returns zero to indicate success. If
(x,y) lies outside the limits of the histogram then the
function returns GSL_EDOM, and none of the bins are modified. The
error handler is not called, since it is often necessary to compute
histograms for a small range of a larger dataset, ignoring any
coordinates outside the range of interest.
This function is similar to gsl_histogram2d_increment but increases
the value of the appropriate bin in the histogram h by the
floating-point number weight.
This function returns the contents of the (i,j)-th bin of the
histogram h. If (i,j) lies outside the valid range of
indices for the histogram then the error handler is called with an error
code of GSL_EDOM and the function returns 0.
These functions find the upper and lower range limits of the i-th
and j-th bins in the x and y directions of the histogram h.
The range limits are stored in xlower and xupper or
ylower and yupper. The lower limits are inclusive
(i.e. events with these coordinates are included in the bin) and the
upper limits are exclusive (i.e. events with the value of the upper
limit are not included and fall in the neighboring higher bin, if it
exists). The functions return 0 to indicate success. If i or
j lies outside the valid range of indices for the histogram then
the error handler is called with an error code of GSL_EDOM.
These functions return the maximum upper and minimum lower range limits
and the number of bins for the x and y directions of the histogram
h. They provide a way of determining these values without
accessing the gsl_histogram2d struct directly.
This function resets all the bins of the histogram h to zero.
Next: Searching 2D histogram ranges, Previous: Copying 2D Histograms, Up: Histograms [Index]
Next: Debye Functions, Previous: Coupling Coefficients, Up: Special Functions [Index]
The Dawson integral is defined by \exp(-x^2) \int_0^x dt \exp(t^2). A table of Dawson’s integral can be found in Abramowitz & Stegun, Table 7.5. The Dawson functions are declared in the header file gsl_sf_dawson.h.
These routines compute the value of Dawson’s integral for x.
Next: Sorting Examples, Previous: Selecting the k smallest or largest elements, Up: Sorting [Index]
The rank of an element is its order in the sorted data. The rank is the inverse of the index permutation, p. It can be computed using the following algorithm,
for (i = 0; i < p->size; i++)
{
size_t pi = p->data[i];
rank->data[pi] = i;
}
This can be computed directly from the function
gsl_permutation_inverse(rank,p).
The following function will print the rank of each element of the vector v,
void
print_rank (gsl_vector * v)
{
size_t i;
size_t n = v->size;
gsl_permutation * perm = gsl_permutation_alloc(n);
gsl_permutation * rank = gsl_permutation_alloc(n);
gsl_sort_vector_index (perm, v);
gsl_permutation_inverse (rank, perm);
for (i = 0; i < n; i++)
{
double vi = gsl_vector_get(v, i);
printf ("element = %d, value = %g, rank = %d\n",
i, vi, rank->data[i]);
}
gsl_permutation_free (perm);
gsl_permutation_free (rank);
}
Next: Combination functions, Previous: Accessing combination elements, Up: Combinations [Index]
This function returns the range (n) of the combination c.
This function returns the number of elements (k) in the combination c.
This function returns a pointer to the array of elements in the combination c.
This function checks that the combination c is valid. The k elements should lie in the range 0 to n-1, with each value occurring once at most and in increasing order.
Previous: Fitting Examples, Up: Least-Squares Fitting [Index]
A summary of formulas and techniques for least squares fitting can be found in the “Statistics” chapter of the Annual Review of Particle Physics prepared by the Particle Data Group,
The Review of Particle Physics is available online at the website given above.
The tests used to prepare these routines are based on the NIST Statistical Reference Datasets. The datasets and their documentation are available from NIST at the following website,
More information on Tikhonov regularization can be found in
The GSL implementation of robust linear regression closely follows the publications
More information about the normal equations and TSQR approach for solving large linear least squares systems can be found in the publications
Previous: Fitting Examples, Up: Least-Squares Fitting [Index]
Next: Nonlinear Least-Squares TRS 2D Subspace, Previous: Nonlinear Least-Squares TRS Dogleg, Up: Nonlinear Least-Squares TRS Overview [Index]
This method is an improvement over the classical dogleg algorithm, which attempts to include information about the Gauss-Newton step while the iteration is still far from the minimum. When the Cauchy point is inside the trust region and the Gauss-Newton point is outside, the method computes a scaled Gauss-Newton point and then takes a dogleg step between the Cauchy point and the scaled Gauss-Newton point. The scaling is calculated to ensure that the reduction in the model m_k is about the same as the reduction provided by the Cauchy point.
gsl-ref-html-2.3/Sparse-Iterative-Solvers.html 0000664 0001750 0001750 00000011235 13055414606 017451 0 ustar edd eddNext: Sparse Linear Algebra Examples, Previous: Overview of Sparse Linear Algebra, Up: Sparse Linear Algebra [Index]
| • Sparse Iterative Solver Overview: | ||
| • Sparse Iterative Solvers Types: | ||
| • Iterating the Sparse Linear System: |
Next: Constructing the knots vector, Previous: Overview of B-splines, Up: Basis Splines [Index]
The computation of B-spline functions requires a preallocated
workspace of type gsl_bspline_workspace.
This function allocates a workspace for computing B-splines of order k. The number of breakpoints is given by nbreak. This leads to n = nbreak + k - 2 basis functions. Cubic B-splines are specified by k = 4. The size of the workspace is O(2k^2 + 5k + nbreak).
This function frees the memory associated with the workspace w.
Next: Irregular Cylindrical Bessel Functions, Up: Bessel Functions [Index]
These routines compute the regular cylindrical Bessel function of zeroth order, J_0(x).
These routines compute the regular cylindrical Bessel function of first order, J_1(x).
These routines compute the regular cylindrical Bessel function of order n, J_n(x).
This routine computes the values of the regular cylindrical Bessel functions J_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
Next: QAG adaptive integration, Previous: Numerical Integration Introduction, Up: Numerical Integration [Index]
The QNG algorithm is a non-adaptive procedure which uses fixed Gauss-Kronrod-Patterson abscissae to sample the integrand at a maximum of 87 points. It is provided for fast integration of smooth functions.
This function applies the Gauss-Kronrod 10-point, 21-point, 43-point and 87-point integration rules in succession until an estimate of the integral of f over (a,b) is achieved within the desired absolute and relative error limits, epsabs and epsrel. The function returns the final approximation, result, an estimate of the absolute error, abserr and the number of function evaluations used, neval. The Gauss-Kronrod rules are designed in such a way that each rule uses all the results of its predecessors, in order to minimize the total number of function evaluations.
Previous: Fitting robust linear regression example, Up: Fitting Examples [Index]
The following program demostrates the large dense linear least squares solvers. This example is adapted from Trefethen and Bau, and fits the function f(t) = \exp{(\sin^3{(10t)}}) on the interval [0,1] with a degree 15 polynomial. The program generates n = 50000 equally spaced points t_i on this interval, calculates the function value and adds random noise to determine the observation value y_i. The entries of the least squares matrix are X_{ij} = t_i^j, representing a polynomial fit. The matrix is highly ill-conditioned, with a condition number of about 1.4 \cdot 10^{11}. The program accumulates the matrix into the least squares system in 5 blocks, each with 10000 rows. This way the full matrix X is never stored in memory. We solve the system with both the normal equations and TSQR methods. The results are shown in the plot below. In the top left plot, we see the unregularized normal equations solution has larger error than TSQR due to the ill-conditioning of the matrix. In the bottom left plot, we show the L-curve, which exhibits multiple corners. In the top right panel, we plot a regularized solution using \lambda = 10^{-6}. The TSQR and normal solutions now agree, however they are unable to provide a good fit due to the damping. This indicates that for some ill-conditioned problems, regularizing the normal equations does not improve the solution. This is further illustrated in the bottom right panel, where we plot the L-curve calculated from the normal equations. The curve agrees with the TSQR curve for larger damping parameters, but for small \lambda, the normal equations approach cannot provide accurate solution vectors leading to numerical inaccuracies in the left portion of the curve.
#include <gsl/gsl_math.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_multifit.h>
#include <gsl/gsl_multilarge.h>
#include <gsl/gsl_blas.h>
/* function to be fitted */
double
func(const double t)
{
double x = sin(10.0 * t);
return exp(x*x*x);
}
/* construct a row of the least squares matrix */
int
build_row(const double t, gsl_vector *row)
{
const size_t p = row->size;
double Xj = 1.0;
size_t j;
for (j = 0; j < p; ++j)
{
gsl_vector_set(row, j, Xj);
Xj *= t;
}
return 0;
}
int
solve_system(const int print_data, const gsl_multilarge_linear_type * T,
const double lambda, const size_t n, const size_t p,
gsl_vector * c)
{
const size_t nblock = 5; /* number of blocks to accumulate */
const size_t nrows = n / nblock; /* number of rows per block */
gsl_multilarge_linear_workspace * w =
gsl_multilarge_linear_alloc(T, p);
gsl_matrix *X = gsl_matrix_alloc(nrows, p);
gsl_vector *y = gsl_vector_alloc(nrows);
gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
const size_t nlcurve = 200;
gsl_vector *reg_param = gsl_vector_alloc(nlcurve);
gsl_vector *rho = gsl_vector_alloc(nlcurve);
gsl_vector *eta = gsl_vector_alloc(nlcurve);
size_t rowidx = 0;
double rnorm, snorm, rcond;
double t = 0.0;
double dt = 1.0 / (n - 1.0);
while (rowidx < n)
{
size_t nleft = n - rowidx; /* number of rows left to accumulate */
size_t nr = GSL_MIN(nrows, nleft); /* number of rows in this block */
gsl_matrix_view Xv = gsl_matrix_submatrix(X, 0, 0, nr, p);
gsl_vector_view yv = gsl_vector_subvector(y, 0, nr);
size_t i;
/* build (X,y) block with 'nr' rows */
for (i = 0; i < nr; ++i)
{
gsl_vector_view row = gsl_matrix_row(&Xv.matrix, i);
double fi = func(t);
double ei = gsl_ran_gaussian (r, 0.1 * fi); /* noise */
double yi = fi + ei;
/* construct this row of LS matrix */
build_row(t, &row.vector);
/* set right hand side value with added noise */
gsl_vector_set(&yv.vector, i, yi);
if (print_data && (i % 100 == 0))
printf("%f %f\n", t, yi);
t += dt;
}
/* accumulate (X,y) block into LS system */
gsl_multilarge_linear_accumulate(&Xv.matrix, &yv.vector, w);
rowidx += nr;
}
if (print_data)
printf("\n\n");
/* compute L-curve */
gsl_multilarge_linear_lcurve(reg_param, rho, eta, w);
/* solve large LS system and store solution in c */
gsl_multilarge_linear_solve(lambda, c, &rnorm, &snorm, w);
/* compute reciprocal condition number */
gsl_multilarge_linear_rcond(&rcond, w);
fprintf(stderr, "=== Method %s ===\n", gsl_multilarge_linear_name(w));
fprintf(stderr, "condition number = %e\n", 1.0 / rcond);
fprintf(stderr, "residual norm = %e\n", rnorm);
fprintf(stderr, "solution norm = %e\n", snorm);
/* output L-curve */
{
size_t i;
for (i = 0; i < nlcurve; ++i)
{
printf("%.12e %.12e %.12e\n",
gsl_vector_get(reg_param, i),
gsl_vector_get(rho, i),
gsl_vector_get(eta, i));
}
printf("\n\n");
}
gsl_matrix_free(X);
gsl_vector_free(y);
gsl_multilarge_linear_free(w);
gsl_rng_free(r);
gsl_vector_free(reg_param);
gsl_vector_free(rho);
gsl_vector_free(eta);
return 0;
}
int
main(int argc, char *argv[])
{
const size_t n = 50000; /* number of observations */
const size_t p = 16; /* polynomial order + 1 */
double lambda = 0.0; /* regularization parameter */
gsl_vector *c_tsqr = gsl_vector_alloc(p);
gsl_vector *c_normal = gsl_vector_alloc(p);
if (argc > 1)
lambda = atof(argv[1]);
/* solve system with TSQR method */
solve_system(1, gsl_multilarge_linear_tsqr, lambda, n, p, c_tsqr);
/* solve system with Normal equations method */
solve_system(0, gsl_multilarge_linear_normal, lambda, n, p, c_normal);
/* output solutions */
{
gsl_vector *v = gsl_vector_alloc(p);
double t;
for (t = 0.0; t <= 1.0; t += 0.01)
{
double f_exact = func(t);
double f_tsqr, f_normal;
build_row(t, v);
gsl_blas_ddot(v, c_tsqr, &f_tsqr);
gsl_blas_ddot(v, c_normal, &f_normal);
printf("%f %e %e %e\n", t, f_exact, f_tsqr, f_normal);
}
gsl_vector_free(v);
}
gsl_vector_free(c_tsqr);
gsl_vector_free(c_normal);
return 0;
}
Previous: Fitting robust linear regression example, Up: Fitting Examples [Index]
Next: Sparse Matrices Accessing Elements, Previous: Sparse Matrices Overview, Up: Sparse Matrices [Index]
The functions for allocating memory for a sparse matrix follow the style of
malloc and free. They also perform their own error checking. If
there is insufficient memory available to allocate a matrix then the functions
call the GSL error handler with an error code of GSL_ENOMEM in addition
to returning a null pointer.
This function allocates a sparse matrix of size n1-by-n2 and
initializes it to all zeros. If the size of the matrix is not known at allocation
time, both n1 and n2 may be set to 1, and they will automatically
grow as elements are added to the matrix. This function sets the
matrix to the triplet representation, which is the easiest for adding
and accessing matrix elements. This function tries to make a reasonable guess
for the number of non-zero elements (nzmax) which will be added to the matrix by
assuming a sparse density of 10\%. The function
gsl_spmatrix_alloc_nzmax can be used if this number is known more
accurately. The workspace is of size O(nzmax).
This function allocates a sparse matrix of size n1-by-n2 and
initializes it to all zeros. If the size of the matrix is not known at allocation
time, both n1 and n2 may be set to 1, and they will automatically
grow as elements are added to the matrix. The parameter nzmax specifies
the maximum number of non-zero elements which will be added to the matrix.
It does not need to be precisely known in advance, since storage space will
automatically grow using gsl_spmatrix_realloc if nzmax is not
large enough. Accurate knowledge of this parameter reduces the number of
reallocation calls required. The parameter sptype specifies the
storage format of the sparse matrix. Possible values are
GSL_SPMATRIX_TRIPLETThis flag specifies triplet storage.
GSL_SPMATRIX_CCSThis flag specifies compressed column storage.
GSL_SPMATRIX_CRSThis flag specifies compressed row storage.
The allocated gsl_spmatrix structure is of size O(nzmax).
This function reallocates the storage space for m to accomodate
nzmax non-zero elements. It is typically called internally by
gsl_spmatrix_set if the user wants to add more elements to the
sparse matrix than the previously specified nzmax.
This function frees the memory associated with the sparse matrix m.
Next: Sparse Matrices Accessing Elements, Previous: Sparse Matrices Overview, Up: Sparse Matrices [Index]
Previous: 6-j Symbols, Up: Coupling Coefficients [Index]
These routines compute the Wigner 9-j coefficient,
{ja jb jc
jd je jf
jg jh ji}
where the arguments are given in half-integer units, ja = two_ja/2, ma = two_ma/2, etc.
The following program illustrates the large nonlinear least squares solvers on a system with significant sparse structure in the Jacobian. The cost function is given by
\Phi(x) &= 1/2 \sum_{i=1}^{p+1} f_i^2
f_i &= \sqrt{\alpha} (x_i - 1), 1 \le i \le p
f_{p+1} &= ||x||^2 - 1/4
with \alpha = 10^{-5}. The residual f_{p+1} imposes a constraint on the p parameters x, to ensure that ||x||^2 \approx {1 \over 4}. The (p+1)-by-p Jacobian for this system is given by
J(x) = [ \sqrt{alpha} I_p; 2 x^T ]
and the normal equations matrix is given by
J^T J = [ \alpha I_p + 4 x x^T ]
Finally, the second directional derivative of f for the geodesic acceleration method is given by
fvv = [ 0; 2 ||v||^2 ]
Since the upper p-by-p block of J is diagonal, this sparse structure should be exploited in the nonlinear solver. For comparison, the following program solves the system for p = 2000 using the dense direct Cholesky solver based on the normal equations matrix J^T J, as well as the iterative Steihaug-Toint solver, based on sparse matrix-vector products J u and J^T u. The program output is shown below.
Method NITER NFEV NJUEV NJTJEV NAEV Init Cost Final cost cond(J) Final |x|^2 Time (s) levenberg-marquardt 25 31 26 26 0 7.1218e+18 1.9555e-02 447.50 2.5044e-01 46.28 levenberg-marquardt+accel 22 23 45 23 22 7.1218e+18 1.9555e-02 447.64 2.5044e-01 33.92 dogleg 37 87 36 36 0 7.1218e+18 1.9555e-02 447.59 2.5044e-01 56.05 double-dogleg 35 88 34 34 0 7.1218e+18 1.9555e-02 447.62 2.5044e-01 52.65 2D-subspace 37 88 36 36 0 7.1218e+18 1.9555e-02 447.71 2.5044e-01 59.75 steihaug-toint 35 88 345 0 0 7.1218e+18 1.9555e-02 inf 2.5044e-01 0.09
The first five rows use methods based on factoring the dense J^T J matrix while the last row uses the iterative Steihaug-Toint method. While the number of Jacobian matrix-vector products (NJUEV) is less for the dense methods, the added time to construct and factor the J^T J matrix (NJTJEV) results in a much larger runtime than the iterative method (see last column).
The program is given below.
#include <stdlib.h>
#include <stdio.h>
#include <sys/time.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_multilarge_nlinear.h>
#include <gsl/gsl_spblas.h>
#include <gsl/gsl_spmatrix.h>
/* parameters for functions */
struct model_params
{
double alpha;
gsl_spmatrix *J;
};
/* penalty function */
int
penalty_f (const gsl_vector * x, void *params, gsl_vector * f)
{
struct model_params *par = (struct model_params *) params;
const double sqrt_alpha = sqrt(par->alpha);
const size_t p = x->size;
size_t i;
double sum = 0.0;
for (i = 0; i < p; ++i)
{
double xi = gsl_vector_get(x, i);
gsl_vector_set(f, i, sqrt_alpha*(xi - 1.0));
sum += xi * xi;
}
gsl_vector_set(f, p, sum - 0.25);
return GSL_SUCCESS;
}
int
penalty_df (CBLAS_TRANSPOSE_t TransJ, const gsl_vector * x,
const gsl_vector * u, void * params, gsl_vector * v,
gsl_matrix * JTJ)
{
struct model_params *par = (struct model_params *) params;
const size_t p = x->size;
size_t j;
/* store 2*x in last row of J */
for (j = 0; j < p; ++j)
{
double xj = gsl_vector_get(x, j);
gsl_spmatrix_set(par->J, p, j, 2.0 * xj);
}
/* compute v = op(J) u */
if (v)
gsl_spblas_dgemv(TransJ, 1.0, par->J, u, 0.0, v);
if (JTJ)
{
gsl_vector_view diag = gsl_matrix_diagonal(JTJ);
/* compute J^T J = [ alpha*I_p + 4 x x^T ] */
gsl_matrix_set_zero(JTJ);
/* store 4 x x^T in lower half of JTJ */
gsl_blas_dsyr(CblasLower, 4.0, x, JTJ);
/* add alpha to diag(JTJ) */
gsl_vector_add_constant(&diag.vector, par->alpha);
}
return GSL_SUCCESS;
}
int
penalty_fvv (const gsl_vector * x, const gsl_vector * v,
void *params, gsl_vector * fvv)
{
const size_t p = x->size;
double normv = gsl_blas_dnrm2(v);
gsl_vector_set_zero(fvv);
gsl_vector_set(fvv, p, 2.0 * normv * normv);
(void)params; /* avoid unused parameter warning */
return GSL_SUCCESS;
}
void
solve_system(const gsl_vector *x0, gsl_multilarge_nlinear_fdf *fdf,
gsl_multilarge_nlinear_parameters *params)
{
const gsl_multilarge_nlinear_type *T = gsl_multilarge_nlinear_trust;
const size_t max_iter = 200;
const double xtol = 1.0e-8;
const double gtol = 1.0e-8;
const double ftol = 1.0e-8;
const size_t n = fdf->n;
const size_t p = fdf->p;
gsl_multilarge_nlinear_workspace *work =
gsl_multilarge_nlinear_alloc(T, params, n, p);
gsl_vector * f = gsl_multilarge_nlinear_residual(work);
gsl_vector * x = gsl_multilarge_nlinear_position(work);
int info;
double chisq0, chisq, rcond, xsq;
struct timeval tv0, tv1;
gettimeofday(&tv0, NULL);
/* initialize solver */
gsl_multilarge_nlinear_init(x0, fdf, work);
/* store initial cost */
gsl_blas_ddot(f, f, &chisq0);
/* iterate until convergence */
gsl_multilarge_nlinear_driver(max_iter, xtol, gtol, ftol,
NULL, NULL, &info, work);
gettimeofday(&tv1, NULL);
/* store final cost */
gsl_blas_ddot(f, f, &chisq);
/* compute final ||x||^2 */
gsl_blas_ddot(x, x, &xsq);
/* store cond(J(x)) */
gsl_multilarge_nlinear_rcond(&rcond, work);
/* print summary */
fprintf(stderr, "%-25s %-5zu %-4zu %-5zu %-6zu %-4zu %-10.4e %-10.4e %-7.2f %-11.4e %.2f\n",
gsl_multilarge_nlinear_trs_name(work),
gsl_multilarge_nlinear_niter(work),
fdf->nevalf,
fdf->nevaldfu,
fdf->nevaldf2,
fdf->nevalfvv,
chisq0,
chisq,
1.0 / rcond,
xsq,
(tv1.tv_sec - tv0.tv_sec) + 1.0e-6 * (tv1.tv_usec - tv0.tv_usec));
gsl_multilarge_nlinear_free(work);
}
int
main (void)
{
const size_t p = 2000;
const size_t n = p + 1;
gsl_vector *f = gsl_vector_alloc(n);
gsl_vector *x = gsl_vector_alloc(p);
/* allocate sparse Jacobian matrix with 2*p non-zero elements in triplet format */
gsl_spmatrix *J = gsl_spmatrix_alloc_nzmax(n, p, 2 * p, GSL_SPMATRIX_TRIPLET);
gsl_multilarge_nlinear_fdf fdf;
gsl_multilarge_nlinear_parameters fdf_params =
gsl_multilarge_nlinear_default_parameters();
struct model_params params;
size_t i;
params.alpha = 1.0e-5;
params.J = J;
/* define function to be minimized */
fdf.f = penalty_f;
fdf.df = penalty_df;
fdf.fvv = penalty_fvv;
fdf.n = n;
fdf.p = p;
fdf.params = ¶ms;
for (i = 0; i < p; ++i)
{
/* starting point */
gsl_vector_set(x, i, i + 1.0);
/* store sqrt(alpha)*I_p in upper p-by-p block of J */
gsl_spmatrix_set(J, i, i, sqrt(params.alpha));
}
fprintf(stderr, "%-25s %-4s %-4s %-5s %-6s %-4s %-10s %-10s %-7s %-11s %-10s\n",
"Method", "NITER", "NFEV", "NJUEV", "NJTJEV", "NAEV", "Init Cost",
"Final cost", "cond(J)", "Final |x|^2", "Time (s)");
fdf_params.scale = gsl_multilarge_nlinear_scale_levenberg;
fdf_params.trs = gsl_multilarge_nlinear_trs_lm;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_lmaccel;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_dogleg;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_ddogleg;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_subspace2D;
solve_system(x, &fdf, &fdf_params);
fdf_params.trs = gsl_multilarge_nlinear_trs_cgst;
solve_system(x, &fdf, &fdf_params);
gsl_vector_free(f);
gsl_vector_free(x);
gsl_spmatrix_free(J);
return 0;
}
Previous: Examples with Simulated Annealing, Up: Simulated Annealing [Index]
Further information is available in the following book,
Next: Bessel Functions, Previous: Special Function Modes, Up: Special Functions [Index]
The Airy functions Ai(x) and Bi(x) are defined by the integral representations,
Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3 + xt) + \sin((1/3) t^3 + xt)) dt
For further information see Abramowitz & Stegun, Section 10.4. The Airy functions are defined in the header file gsl_sf_airy.h.
| • Airy Functions: | ||
| • Derivatives of Airy Functions: | ||
| • Zeros of Airy Functions: | ||
| • Zeros of Derivatives of Airy Functions: |
Next: Multiset properties, Previous: Multiset allocation, Up: Multisets [Index]
The following function can be used to access the elements of a multiset.
This function returns the value of the i-th element of the
multiset c. If i lies outside the allowed range of 0 to
k-1 then the error handler is invoked and 0 is returned. An inline version of this function is used when HAVE_INLINE is defined.
Next: Selecting the k smallest or largest elements, Previous: Sorting objects, Up: Sorting [Index]
The following functions will sort the elements of an array or vector,
either directly or indirectly. They are defined for all real and integer
types using the normal suffix rules. For example, the float
versions of the array functions are gsl_sort_float and
gsl_sort_float_index. The corresponding vector functions are
gsl_sort_vector_float and gsl_sort_vector_float_index. The
prototypes are available in the header files gsl_sort_float.h
gsl_sort_vector_float.h. The complete set of prototypes can be
included using the header files gsl_sort.h and
gsl_sort_vector.h.
There are no functions for sorting complex arrays or vectors, since the ordering of complex numbers is not uniquely defined. To sort a complex vector by magnitude compute a real vector containing the magnitudes of the complex elements, and sort this vector indirectly. The resulting index gives the appropriate ordering of the original complex vector.
This function sorts the n elements of the array data with stride stride into ascending numerical order.
This function sorts the n elements of the array data1 with stride stride1 into ascending numerical order, while making the same rearrangement of the array data2 with stride stride2, also of size n.
This function sorts the elements of the vector v into ascending numerical order.
This function sorts the elements of the vector v1 into ascending numerical order, while making the same rearrangement of the vector v2.
This function indirectly sorts the n elements of the array data with stride stride into ascending order, storing the resulting permutation in p. The array p must be allocated with a sufficient length to store the n elements of the permutation. The elements of p give the index of the array element which would have been stored in that position if the array had been sorted in place. The array data is not changed.
This function indirectly sorts the elements of the vector v into ascending order, storing the resulting permutation in p. The elements of p give the index of the vector element which would have been stored in that position if the vector had been sorted in place. The first element of p gives the index of the least element in v, and the last element of p gives the index of the greatest element in v. The vector v is not changed.
Next: Selecting the k smallest or largest elements, Previous: Sorting objects, Up: Sorting [Index]
The library is installed as a single file, libgsl.a. A shared version of the library libgsl.so is also installed on systems that support shared libraries. The default location of these files is /usr/local/lib. If this directory is not on the standard search path of your linker you will also need to provide its location as a command line flag.
To link against the library you need to specify both the main library and a supporting CBLAS library, which provides standard basic linear algebra subroutines. A suitable CBLAS implementation is provided in the library libgslcblas.a if your system does not provide one. The following example shows how to link an application with the library,
$ gcc -L/usr/local/lib example.o -lgsl -lgslcblas -lm
The default library path for gcc searches /usr/local/lib
automatically so the -L option can be omitted when GSL is
installed in its default location.
The option -lm links with the system math library. On some
systems it is not needed.3
For a tutorial introduction to the GNU C Compiler and related programs, see An Introduction to GCC (ISBN 0954161793).4
gsl-ref-html-2.3/Sparse-Linear-Algebra-References-and-Further-Reading.html 0000664 0001750 0001750 00000010072 13055414606 024430 0 ustar edd eddPrevious: Sparse Linear Algebra Examples, Up: Sparse Linear Algebra [Index]
The implementation of the GMRES iterative solver closely follows the publications
Previous: Linear regression with a constant term, Up: Linear regression [Index]
The functions described in this section can be used to perform least-squares fits to a straight line model without a constant term, Y = c_1 X.
This function computes the best-fit linear regression coefficient c1 of the model Y = c_1 X for the datasets (x, y), two vectors of length n with strides xstride and ystride. The errors on y are assumed unknown so the variance of the parameter c1 is estimated from the scatter of the points around the best-fit line and returned via the parameter cov11. The sum of squares of the residuals from the best-fit line is returned in sumsq.
This function computes the best-fit linear regression coefficient c1 of the model Y = c_1 X for the weighted datasets (x, y), two vectors of length n with strides xstride and ystride. The vector w, of length n and stride wstride, specifies the weight of each datapoint. The weight is the reciprocal of the variance for each datapoint in y.
The variance of the parameter c1 is computed using the weights and returned via the parameter cov11. The weighted sum of squares of the residuals from the best-fit line, \chi^2, is returned in chisq.
This function uses the best-fit linear regression coefficient c1 and its covariance cov11 to compute the fitted function y and its standard deviation y_err for the model Y = c_1 X at the point x.
Previous: Linear regression with a constant term, Up: Linear regression [Index]
Next: The Beta Distribution, Previous: The F-distribution, Up: Random Number Distributions [Index]
The t-distribution arises in statistics. If Y_1 has a normal distribution and Y_2 has a chi-squared distribution with \nu degrees of freedom then the ratio,
X = { Y_1 \over \sqrt{Y_2 / \nu} }
has a t-distribution t(x;\nu) with \nu degrees of freedom.
This function returns a random variate from the t-distribution. The distribution function is,
p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)}
(1 + x^2/\nu)^{-(\nu + 1)/2} dx
for -\infty < x < +\infty.
This function computes the probability density p(x) at x for a t-distribution with nu degrees of freedom, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the t-distribution with nu degrees of freedom.
Next: Volume Area and Length, Previous: Speed and Nautical Units, Up: Physical Constants [Index]
GSL_CONST_MKSA_POINTThe length of 1 printer’s point (1/72 inch).
GSL_CONST_MKSA_TEXPOINTThe length of 1 TeX point (1/72.27 inch).
Next: Exchanging rows and columns, Previous: Copying matrices, Up: Matrices [Index]
The functions described in this section copy a row or column of a matrix
into a vector. This allows the elements of the vector and the matrix to
be modified independently. Note that if the matrix and the vector point
to overlapping regions of memory then the result will be undefined. The
same effect can be achieved with more generality using
gsl_vector_memcpy with vector views of rows and columns.
This function copies the elements of the i-th row of the matrix m into the vector v. The length of the vector must be the same as the length of the row.
This function copies the elements of the j-th column of the matrix m into the vector v. The length of the vector must be the same as the length of the column.
This function copies the elements of the vector v into the i-th row of the matrix m. The length of the vector must be the same as the length of the row.
This function copies the elements of the vector v into the j-th column of the matrix m. The length of the vector must be the same as the length of the column.
Next: Coulomb Wave Functions, Up: Coulomb Functions [Index]
These routines compute the lowest-order normalized hydrogenic bound state radial wavefunction R_1 := 2Z \sqrt{Z} \exp(-Z r).
These routines compute the n-th normalized hydrogenic bound state radial wavefunction,
R_n := 2 (Z^{3/2}/n^2) \sqrt{(n-l-1)!/(n+l)!} \exp(-Z r/n) (2Zr/n)^l
L^{2l+1}_{n-l-1}(2Zr/n).
where L^a_b(x) is the generalized Laguerre polynomial (see Laguerre Functions). The normalization is chosen such that the wavefunction \psi is given by \psi(n,l,r) = R_n Y_{lm}.
Previous: 2D Higher-level Interface, Up: Interpolation [Index]
The following example performs bilinear interpolation on the unit square, using z values of (0,1,0.5,1) going clockwise around the square.
#include <stdio.h>
#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_interp2d.h>
#include <gsl/gsl_spline2d.h>
int
main()
{
const gsl_interp2d_type *T = gsl_interp2d_bilinear;
const size_t N = 100; /* number of points to interpolate */
const double xa[] = { 0.0, 1.0 }; /* define unit square */
const double ya[] = { 0.0, 1.0 };
const size_t nx = sizeof(xa) / sizeof(double); /* x grid points */
const size_t ny = sizeof(ya) / sizeof(double); /* y grid points */
double *za = malloc(nx * ny * sizeof(double));
gsl_spline2d *spline = gsl_spline2d_alloc(T, nx, ny);
gsl_interp_accel *xacc = gsl_interp_accel_alloc();
gsl_interp_accel *yacc = gsl_interp_accel_alloc();
size_t i, j;
/* set z grid values */
gsl_spline2d_set(spline, za, 0, 0, 0.0);
gsl_spline2d_set(spline, za, 0, 1, 1.0);
gsl_spline2d_set(spline, za, 1, 1, 0.5);
gsl_spline2d_set(spline, za, 1, 0, 1.0);
/* initialize interpolation */
gsl_spline2d_init(spline, xa, ya, za, nx, ny);
/* interpolate N values in x and y and print out grid for plotting */
for (i = 0; i < N; ++i)
{
double xi = i / (N - 1.0);
for (j = 0; j < N; ++j)
{
double yj = j / (N - 1.0);
double zij = gsl_spline2d_eval(spline, xi, yj, xacc, yacc);
printf("%f %f %f\n", xi, yj, zij);
}
printf("\n");
}
gsl_spline2d_free(spline);
gsl_interp_accel_free(xacc);
gsl_interp_accel_free(yacc);
free(za);
return 0;
}
The results of the interpolation are shown in the following plot, where the corners are labeled with their fixed z values.
gsl-ref-html-2.3/Special-Function-Modes.html 0000664 0001750 0001750 00000011047 13055414560 017037 0 ustar edd eddNext: Airy Functions and Derivatives, Previous: The gsl_sf_result struct, Up: Special Functions [Index]
The goal of the library is to achieve double precision accuracy wherever
possible. However the cost of evaluating some special functions to
double precision can be significant, particularly where very high order
terms are required. In these cases a mode argument allows the
accuracy of the function to be reduced in order to improve performance.
The following precision levels are available for the mode argument,
GSL_PREC_DOUBLEDouble-precision, a relative accuracy of approximately 2 * 10^-16.
GSL_PREC_SINGLESingle-precision, a relative accuracy of approximately 10^-7.
GSL_PREC_APPROXApproximate values, a relative accuracy of approximately 5 * 10^-4.
The approximate mode provides the fastest evaluation at the lowest accuracy.
gsl-ref-html-2.3/QAGI-adaptive-integration-on-infinite-intervals.html 0000664 0001750 0001750 00000015322 13055414453 023707 0 ustar edd eddNext: QAWC adaptive integration for Cauchy principal values, Previous: QAGP adaptive integration with known singular points, Up: Numerical Integration [Index]
This function computes the integral of the function f over the infinite interval (-\infty,+\infty). The integral is mapped onto the semi-open interval (0,1] using the transformation x = (1-t)/t,
\int_{-\infty}^{+\infty} dx f(x) =
\int_0^1 dt (f((1-t)/t) + f((-1+t)/t))/t^2.
It is then integrated using the QAGS algorithm. The normal 21-point Gauss-Kronrod rule of QAGS is replaced by a 15-point rule, because the transformation can generate an integrable singularity at the origin. In this case a lower-order rule is more efficient.
This function computes the integral of the function f over the semi-infinite interval (a,+\infty). The integral is mapped onto the semi-open interval (0,1] using the transformation x = a + (1-t)/t,
\int_{a}^{+\infty} dx f(x) =
\int_0^1 dt f(a + (1-t)/t)/t^2
and then integrated using the QAGS algorithm.
This function computes the integral of the function f over the semi-infinite interval (-\infty,b). The integral is mapped onto the semi-open interval (0,1] using the transformation x = b - (1-t)/t,
\int_{-\infty}^{b} dx f(x) =
\int_0^1 dt f(b - (1-t)/t)/t^2
and then integrated using the QAGS algorithm.
Next: The Pascal Distribution, Previous: The Multinomial Distribution, Up: Random Number Distributions [Index]
This function returns a random integer from the negative binomial distribution, the number of failures occurring before n successes in independent trials with probability p of success. The probability distribution for negative binomial variates is,
p(k) = {\Gamma(n + k) \over \Gamma(k+1) \Gamma(n) } p^n (1-p)^k
Note that n is not required to be an integer.
This function computes the probability p(k) of obtaining k from a negative binomial distribution with parameters p and n, using the formula given above.
These functions compute the cumulative distribution functions P(k), Q(k) for the negative binomial distribution with parameters p and n.
Next: DWT Examples, Previous: DWT Initialization, Up: Wavelet Transforms [Index]
This sections describes the actual functions performing the discrete wavelet transform. Note that the transforms use periodic boundary conditions. If the signal is not periodic in the sample length then spurious coefficients will appear at the beginning and end of each level of the transform.
| • DWT in one dimension: | ||
| • DWT in two dimension: |
Next: The Hypergeometric Distribution, Previous: The Pascal Distribution, Up: Random Number Distributions [Index]
This function returns a random integer from the geometric distribution, the number of independent trials with probability p until the first success. The probability distribution for geometric variates is,
p(k) = p (1-p)^(k-1)
for k >= 1. Note that the distribution begins with k=1 with this definition. There is another convention in which the exponent k-1 is replaced by k.
This function computes the probability p(k) of obtaining k from a geometric distribution with probability parameter p, using the formula given above.
These functions compute the cumulative distribution functions P(k), Q(k) for the geometric distribution with parameter p.
Next: Eta Function, Previous: Riemann Zeta Function Minus One, Up: Zeta Functions [Index]
The Hurwitz zeta function is defined by \zeta(s,q) = \sum_0^\infty (k+q)^{-s}.
These routines compute the Hurwitz zeta function \zeta(s,q) for s > 1, q > 0.
Previous: Eigenvalue and Eigenvector Examples, Up: Eigensystems [Index]
Further information on the algorithms described in this section can be found in the following book,
Further information on the generalized eigensystems QZ algorithm can be found in this paper,
Eigensystem routines for very large matrices can be found in the Fortran library LAPACK. The LAPACK library is described in,
The LAPACK source code can be found at the website above along with an online copy of the users guide.
gsl-ref-html-2.3/Acceleration-functions-without-error-estimation.html 0000664 0001750 0001750 00000016207 13055414545 024236 0 ustar edd eddNext: Example of accelerating a series, Previous: Acceleration functions, Up: Series Acceleration [Index]
The functions described in this section compute the Levin u-transform of series and attempt to estimate the error from the “truncation error” in the extrapolation, the difference between the final two approximations. Using this method avoids the need to compute an intermediate table of derivatives because the error is estimated from the behavior of the extrapolated value itself. Consequently this algorithm is an O(N) process and only requires O(N) terms of storage. If the series converges sufficiently fast then this procedure can be acceptable. It is appropriate to use this method when there is a need to compute many extrapolations of series with similar convergence properties at high-speed. For example, when numerically integrating a function defined by a parameterized series where the parameter varies only slightly. A reliable error estimate should be computed first using the full algorithm described above in order to verify the consistency of the results.
This function allocates a workspace for a Levin u-transform of n terms, without error estimation. The size of the workspace is O(3n).
This function frees the memory associated with the workspace w.
This function takes the terms of a series in array of size
array_size and computes the extrapolated limit of the series using
a Levin u-transform. Additional working space must be provided in
w. The extrapolated sum is stored in sum_accel. The actual
term-by-term sum is returned in w->sum_plain. The algorithm
terminates when the difference between two successive extrapolations
reaches a minimum or is sufficiently small. The difference between these
two values is used as estimate of the error and is stored in
abserr_trunc. To improve the reliability of the algorithm the
extrapolated values are replaced by moving averages when calculating the
truncation error, smoothing out any fluctuations.
Next: Example of accelerating a series, Previous: Acceleration functions, Up: Series Acceleration [Index]
Next: Restriction Functions, Previous: Hyperbolic Trigonometric Functions, Up: Trigonometric Functions [Index]
This function converts the polar coordinates (r,theta) to rectilinear coordinates (x,y), x = r\cos(\theta), y = r\sin(\theta).
This function converts the rectilinear coordinates (x,y) to polar coordinates (r,theta), such that x = r\cos(\theta), y = r\sin(\theta). The argument theta lies in the range [-\pi, \pi].
Next: The Negative Binomial Distribution, Previous: The Binomial Distribution, Up: Random Number Distributions [Index]
This function computes a random sample n[] from the multinomial distribution formed by N trials from an underlying distribution p[K]. The distribution function for n[] is,
P(n_1, n_2, ..., n_K) = (N!/(n_1! n_2! ... n_K!)) p_1^n_1 p_2^n_2 ... p_K^n_K
where (n_1, n_2, ..., n_K) are nonnegative integers with sum_{k=1}^K n_k = N, and (p_1, p_2, ..., p_K) is a probability distribution with \sum p_i = 1. If the array p[K] is not normalized then its entries will be treated as weights and normalized appropriately. The arrays n[] and p[] must both be of length K.
Random variates are generated using the conditional binomial method (see C.S. Davis, The computer generation of multinomial random variates, Comp. Stat. Data Anal. 16 (1993) 205–217 for details).
This function computes the probability P(n_1, n_2, ..., n_K) of sampling n[K] from a multinomial distribution with parameters p[K], using the formula given above.
This function returns the logarithm of the probability for the multinomial distribution P(n_1, n_2, ..., n_K) with parameters p[K].
Next: Power Function, Previous: Logarithm and Related Functions, Up: Special Functions [Index]
The routines described in this section compute the angular and radial Mathieu functions, and their characteristic values. Mathieu functions are the solutions of the following two differential equations:
d^2y/dv^2 + (a - 2q\cos 2v)y = 0 d^2f/du^2 - (a - 2q\cosh 2u)f = 0
The angular Mathieu functions ce_r(x,q), se_r(x,q) are the even and odd periodic solutions of the first equation, which is known as Mathieu’s equation. These exist only for the discrete sequence of characteristic values a=a_r(q) (even-periodic) and a=b_r(q) (odd-periodic).
The radial Mathieu functions Mc^{(j)}_{r}(z,q), Ms^{(j)}_{r}(z,q) are the solutions of the second equation, which is referred to as Mathieu’s modified equation. The radial Mathieu functions of the first, second, third and fourth kind are denoted by the parameter j, which takes the value 1, 2, 3 or 4.
For more information on the Mathieu functions, see Abramowitz and Stegun, Chapter 20. These functions are defined in the header file gsl_sf_mathieu.h.
| • Mathieu Function Workspace: | ||
| • Mathieu Function Characteristic Values: | ||
| • Angular Mathieu Functions: | ||
| • Radial Mathieu Functions: |
Next: Fitting Examples, Previous: Large Dense Linear Systems, Up: Least-Squares Fitting [Index]
When using models based on polynomials, care should be taken when constructing the design matrix X. If the x values are large, then the matrix X could be ill-conditioned since its columns are powers of x, leading to unstable least-squares solutions. In this case it can often help to center and scale the x values using the mean and standard deviation:
x' = (x - mu)/sigma
and then construct the X matrix using the transformed values x'.
gsl-ref-html-2.3/Root-Finding-Caveats.html 0000664 0001750 0001750 00000013443 13055414601 016510 0 ustar edd eddNext: Initializing the Solver, Previous: Root Finding Overview, Up: One dimensional Root-Finding [Index]
Note that root finding functions can only search for one root at a time. When there are several roots in the search area, the first root to be found will be returned; however it is difficult to predict which of the roots this will be. In most cases, no error will be reported if you try to find a root in an area where there is more than one.
Care must be taken when a function may have a multiple root (such as f(x) = (x-x_0)^2 or f(x) = (x-x_0)^3). It is not possible to use root-bracketing algorithms on even-multiplicity roots. For these algorithms the initial interval must contain a zero-crossing, where the function is negative at one end of the interval and positive at the other end. Roots with even-multiplicity do not cross zero, but only touch it instantaneously. Algorithms based on root bracketing will still work for odd-multiplicity roots (e.g. cubic, quintic, …). Root polishing algorithms generally work with higher multiplicity roots, but at a reduced rate of convergence. In these cases the Steffenson algorithm can be used to accelerate the convergence of multiple roots.
While it is not absolutely required that f have a root within the search region, numerical root finding functions should not be used haphazardly to check for the existence of roots. There are better ways to do this. Because it is easy to create situations where numerical root finders can fail, it is a bad idea to throw a root finder at a function you do not know much about. In general it is best to examine the function visually by plotting before searching for a root.
Next: Initializing the Solver, Previous: Root Finding Overview, Up: One dimensional Root-Finding [Index]
Next: Sparse Matrices Copying, Previous: Sparse Matrices Initializing Elements, Up: Sparse Matrices [Index]
This function writes the elements of the matrix m to the stream
stream in binary format. The return value is 0 for success and
GSL_EFAILED if there was a problem writing to the file. Since the
data is written in the native binary format it may not be portable
between different architectures.
This function reads into the matrix m from the open stream
stream in binary format. The matrix m must be preallocated
with the correct storage format, dimensions and have a sufficiently large nzmax
in order to read in all matrix elements, otherwise GSL_EBADLEN
is returned. The return value is 0 for success and
GSL_EFAILED if there was a problem reading from the file. The
data is assumed to have been written in the native binary format on the
same architecture.
This function writes the elements of the matrix m line-by-line to
the stream stream using the format specifier format, which
should be one of the %g, %e or %f formats for
floating point numbers. The function returns 0 for success and
GSL_EFAILED if there was a problem writing to the file. The
input matrix m may be in any storage format, and the output file
will be written in MatrixMarket format.
This function reads sparse matrix data in the MatrixMarket format
from the stream stream and stores it in a newly allocated matrix
which is returned in triplet format. The function returns 0 for success and
GSL_EFAILED if there was a problem reading from the file. The
user should free the returned matrix when it is no longer needed.
Next: Sparse Matrices Copying, Previous: Sparse Matrices Initializing Elements, Up: Sparse Matrices [Index]
Next: Type Index, Previous: Function Index, Up: Top [Index]
| Jump to: | A D E F G H I M O S T V |
|---|
| Jump to: | A D E F G H I M O S T V |
|---|
Next: Type Index, Previous: Function Index, Up: Top [Index]
Next: Modified Cholesky Decomposition, Previous: Cholesky Decomposition, Up: Linear Algebra [Index]
A symmetric, positive definite square matrix A has an alternate Cholesky decomposition into a product of a lower unit triangular matrix L, a diagonal matrix D and L^T, given by L D L^T. This is equivalent to the Cholesky formulation discussed above, with the standard Cholesky lower triangular factor given by L D^{1 \over 2}. For ill-conditioned matrices, it can help to use a pivoting strategy to prevent the entries of D and L from growing too large, and also ensure D_1 \ge D_2 \ge \cdots \ge D_n > 0, where D_i are the diagonal entries of D. The final decomposition is given by
P A P^T = L D L^T
where P is a permutation matrix.
This function factors the symmetric, positive-definite square matrix A into the Pivoted Cholesky decomposition P A P^T = L D L^T. On input, the values from the diagonal and lower-triangular part of the matrix A are used to construct the factorization. On output the diagonal of the input matrix A stores the diagonal elements of D, and the lower triangular portion of A contains the matrix L. Since L has ones on its diagonal these do not need to be explicitely stored. The upper triangular portion of A is unmodified. The permutation matrix P is stored in p on output.
This function solves the system A x = b using the Pivoted Cholesky
decomposition of A held in the matrix LDLT and permutation
p which must have been previously computed by gsl_linalg_pcholesky_decomp.
This function solves the system A x = b in-place using the Pivoted Cholesky
decomposition of A held in the matrix LDLT and permutation
p which must have been previously computed by gsl_linalg_pcholesky_decomp.
On input, x contains the right hand side vector b which is
replaced by the solution vector on output.
This function computes the pivoted Cholesky factorization of the matrix S A S, where the input matrix A is symmetric and positive definite, and the diagonal scaling matrix S is computed to reduce the condition number of A as much as possible. See Cholesky Decomposition for more information on the matrix S. The Pivoted Cholesky decomposition satisfies P S A S P^T = L D L^T. On input, the values from the diagonal and lower-triangular part of the matrix A are used to construct the factorization. On output the diagonal of the input matrix A stores the diagonal elements of D, and the lower triangular portion of A contains the matrix L. Since L has ones on its diagonal these do not need to be explicitely stored. The upper triangular portion of A is unmodified. The permutation matrix P is stored in p on output. The diagonal scaling transformation is stored in S on output.
This function solves the system (S A S) (S^{-1} x) = S b using the Pivoted Cholesky
decomposition of S A S held in the matrix LDLT, permutation
p, and vector S, which must have been previously computed by
gsl_linalg_pcholesky_decomp2.
This function solves the system (S A S) (S^{-1} x) = S b in-place using the Pivoted Cholesky
decomposition of S A S held in the matrix LDLT, permutation
p and vector S, which must have been previously computed by
gsl_linalg_pcholesky_decomp2.
On input, x contains the right hand side vector b which is
replaced by the solution vector on output.
This function computes the inverse of the matrix A, using the Pivoted Cholesky decomposition stored in LDLT and p. On output, the matrix Ainv contains A^{-1}.
This function estimates the reciprocal condition number (using the 1-norm) of the symmetric positive definite matrix A, using its pivoted Cholesky decomposition provided in LDLT. The reciprocal condition number estimate, defined as 1 / (||A||_1 \cdot ||A^{-1}||_1), is stored in rcond. Additional workspace of size 3 N is required in work.
Next: Modified Cholesky Decomposition, Previous: Cholesky Decomposition, Up: Linear Algebra [Index]
Next: Iteration of the multidimensional solver, Previous: Initializing the Multidimensional Solver, Up: Multidimensional Root-Finding [Index]
You must provide n functions of n variables for the root finders to operate on. In order to allow for general parameters the functions are defined by the following data types:
This data type defines a general system of functions with parameters.
int (* f) (const gsl_vector * x, void * params, gsl_vector * f)this function should store the vector result f(x,params) in f for argument x and parameters params, returning an appropriate error code if the function cannot be computed.
size_t nthe dimension of the system, i.e. the number of components of the vectors x and f.
void * paramsa pointer to the parameters of the function.
Here is an example using Powell’s test function,
f_1(x) = A x_0 x_1 - 1, f_2(x) = exp(-x_0) + exp(-x_1) - (1 + 1/A)
with A = 10^4. The following code defines a
gsl_multiroot_function system F which you could pass to a
solver:
struct powell_params { double A; };
int
powell (gsl_vector * x, void * p, gsl_vector * f) {
struct powell_params * params
= (struct powell_params *)p;
const double A = (params->A);
const double x0 = gsl_vector_get(x,0);
const double x1 = gsl_vector_get(x,1);
gsl_vector_set (f, 0, A * x0 * x1 - 1);
gsl_vector_set (f, 1, (exp(-x0) + exp(-x1)
- (1.0 + 1.0/A)));
return GSL_SUCCESS
}
gsl_multiroot_function F;
struct powell_params params = { 10000.0 };
F.f = &powell;
F.n = 2;
F.params = ¶ms;
This data type defines a general system of functions with parameters and the corresponding Jacobian matrix of derivatives,
int (* f) (const gsl_vector * x, void * params, gsl_vector * f)this function should store the vector result f(x,params) in f for argument x and parameters params, returning an appropriate error code if the function cannot be computed.
int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)this function should store the n-by-n matrix result J_ij = d f_i(x,params) / d x_j in J for argument x and parameters params, returning an appropriate error code if the function cannot be computed.
int (* fdf) (const gsl_vector * x, void * params, gsl_vector * f, gsl_matrix * J)This function should set the values of the f and J as above, for arguments x and parameters params. This function provides an optimization of the separate functions for f(x) and J(x)—it is always faster to compute the function and its derivative at the same time.
size_t nthe dimension of the system, i.e. the number of components of the vectors x and f.
void * paramsa pointer to the parameters of the function.
The example of Powell’s test function defined above can be extended to include analytic derivatives using the following code,
int
powell_df (gsl_vector * x, void * p, gsl_matrix * J)
{
struct powell_params * params
= (struct powell_params *)p;
const double A = (params->A);
const double x0 = gsl_vector_get(x,0);
const double x1 = gsl_vector_get(x,1);
gsl_matrix_set (J, 0, 0, A * x1);
gsl_matrix_set (J, 0, 1, A * x0);
gsl_matrix_set (J, 1, 0, -exp(-x0));
gsl_matrix_set (J, 1, 1, -exp(-x1));
return GSL_SUCCESS
}
int
powell_fdf (gsl_vector * x, void * p,
gsl_matrix * f, gsl_matrix * J) {
struct powell_params * params
= (struct powell_params *)p;
const double A = (params->A);
const double x0 = gsl_vector_get(x,0);
const double x1 = gsl_vector_get(x,1);
const double u0 = exp(-x0);
const double u1 = exp(-x1);
gsl_vector_set (f, 0, A * x0 * x1 - 1);
gsl_vector_set (f, 1, u0 + u1 - (1 + 1/A));
gsl_matrix_set (J, 0, 0, A * x1);
gsl_matrix_set (J, 0, 1, A * x0);
gsl_matrix_set (J, 1, 0, -u0);
gsl_matrix_set (J, 1, 1, -u1);
return GSL_SUCCESS
}
gsl_multiroot_function_fdf FDF;
FDF.f = &powell_f;
FDF.df = &powell_df;
FDF.fdf = &powell_fdf;
FDF.n = 2;
FDF.params = 0;
Note that the function powell_fdf is able to reuse existing terms
from the function when calculating the Jacobian, thus saving time.
Next: Iteration of the multidimensional solver, Previous: Initializing the Multidimensional Solver, Up: Multidimensional Root-Finding [Index]
Next: Shuffling and Sampling, Previous: The Hypergeometric Distribution, Up: Random Number Distributions [Index]
This function returns a random integer from the logarithmic distribution. The probability distribution for logarithmic random variates is,
p(k) = {-1 \over \log(1-p)} {(p^k \over k)}
for k >= 1.
This function computes the probability p(k) of obtaining k from a logarithmic distribution with probability parameter p, using the formula given above.
Next: Autocorrelation, Previous: Absolute deviation, Up: Statistics [Index]
This function computes the skewness of data, a dataset of length n with stride stride. The skewness is defined as,
skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3
where x_i are the elements of the dataset data. The skewness measures the asymmetry of the tails of a distribution.
The function computes the mean and estimated standard deviation of
data via calls to gsl_stats_mean and gsl_stats_sd.
This function computes the skewness of the dataset data using the given values of the mean mean and standard deviation sd,
skew = (1/N) \sum ((x_i - mean)/sd)^3
These functions are useful if you have already computed the mean and standard deviation of data and want to avoid recomputing them.
This function computes the kurtosis of data, a dataset of length n with stride stride. The kurtosis is defined as,
kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4) - 3
The kurtosis measures how sharply peaked a distribution is, relative to its width. The kurtosis is normalized to zero for a Gaussian distribution.
This function computes the kurtosis of the dataset data using the given values of the mean mean and standard deviation sd,
kurtosis = ((1/N) \sum ((x_i - mean)/sd)^4) - 3
This function is useful if you have already computed the mean and standard deviation of data and want to avoid recomputing them.
Next: Autocorrelation, Previous: Absolute deviation, Up: Statistics [Index]
Next: Basis Splines, Previous: Least-Squares Fitting, Up: Top [Index]
This chapter describes functions for multidimensional nonlinear least-squares fitting. There are generally two classes of algorithms for solving nonlinear least squares problems, which fall under line search methods and trust region methods. GSL currently implements only trust region methods and provides the user with full access to intermediate steps of the iteration. The user also has the ability to tune a number of parameters which affect low-level aspects of the algorithm which can help to accelerate convergence for the specific problem at hand. GSL provides two separate interfaces for nonlinear least squares fitting. The first is designed for small to moderate sized problems, and the second is designed for very large problems, which may or may not have significant sparse structure.
The header file gsl_multifit_nlinear.h contains prototypes for the multidimensional nonlinear fitting functions and related declarations relating to the small to moderate sized systems.
The header file gsl_multilarge_nlinear.h contains prototypes for the multidimensional nonlinear fitting functions and related declarations relating to large systems.
Next: Basis Splines, Previous: Least-Squares Fitting, Up: Top [Index]
This function computes the numerical derivative of the function f at the point x using an adaptive central difference algorithm with a step-size of h. The derivative is returned in result and an estimate of its absolute error is returned in abserr.
The initial value of h is used to estimate an optimal step-size, based on the scaling of the truncation error and round-off error in the derivative calculation. The derivative is computed using a 5-point rule for equally spaced abscissae at x-h, x-h/2, x, x+h/2, x+h, with an error estimate taken from the difference between the 5-point rule and the corresponding 3-point rule x-h, x, x+h. Note that the value of the function at x does not contribute to the derivative calculation, so only 4-points are actually used.
This function computes the numerical derivative of the function f at the point x using an adaptive forward difference algorithm with a step-size of h. The function is evaluated only at points greater than x, and never at x itself. The derivative is returned in result and an estimate of its absolute error is returned in abserr. This function should be used if f(x) has a discontinuity at x, or is undefined for values less than x.
The initial value of h is used to estimate an optimal step-size, based on the scaling of the truncation error and round-off error in the derivative calculation. The derivative at x is computed using an “open” 4-point rule for equally spaced abscissae at x+h/4, x+h/2, x+3h/4, x+h, with an error estimate taken from the difference between the 4-point rule and the corresponding 2-point rule x+h/2, x+h.
This function computes the numerical derivative of the function f at the point x using an adaptive backward difference algorithm with a step-size of h. The function is evaluated only at points less than x, and never at x itself. The derivative is returned in result and an estimate of its absolute error is returned in abserr. This function should be used if f(x) has a discontinuity at x, or is undefined for values greater than x.
This function is equivalent to calling gsl_deriv_forward with a
negative step-size.
Next: Sparse Linear Algebra References and Further Reading, Previous: Sparse Iterative Solvers, Up: Sparse Linear Algebra [Index]
This example program demonstrates the sparse linear algebra routines on the solution of a simple 1D Poisson equation on [0,1]:
u''(x) = f(x) = -\pi^2 \sin(\pi x)
with boundary conditions u(0) = u(1) = 0. The analytic solution of this simple problem is u(x) = \sin{\pi x}. We will solve this problem by finite differencing the left hand side to give
1/h^2 ( u_(i+1) - 2 u_i + u_(i-1) ) = f_i
Defining a grid of N points, h = 1/(N-1). In the finite difference equation above, u_0 = u_{N-1} = 0 are known from the boundary conditions, so we will only put the equations for i = 1, ..., N-2 into the matrix system. The resulting (N-2) \times (N-2) matrix equation is An example program which constructs and solves this system is given below. The system is solved using the iterative GMRES solver. Here is the output of the program:
iter 0 residual = 4.297275996844e-11 Converged
showing that the method converged in a single iteration. The calculated solution is shown in the following plot.
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_spmatrix.h>
#include <gsl/gsl_splinalg.h>
int
main()
{
const size_t N = 100; /* number of grid points */
const size_t n = N - 2; /* subtract 2 to exclude boundaries */
const double h = 1.0 / (N - 1.0); /* grid spacing */
gsl_spmatrix *A = gsl_spmatrix_alloc(n ,n); /* triplet format */
gsl_spmatrix *C; /* compressed format */
gsl_vector *f = gsl_vector_alloc(n); /* right hand side vector */
gsl_vector *u = gsl_vector_alloc(n); /* solution vector */
size_t i;
/* construct the sparse matrix for the finite difference equation */
/* construct first row */
gsl_spmatrix_set(A, 0, 0, -2.0);
gsl_spmatrix_set(A, 0, 1, 1.0);
/* construct rows [1:n-2] */
for (i = 1; i < n - 1; ++i)
{
gsl_spmatrix_set(A, i, i + 1, 1.0);
gsl_spmatrix_set(A, i, i, -2.0);
gsl_spmatrix_set(A, i, i - 1, 1.0);
}
/* construct last row */
gsl_spmatrix_set(A, n - 1, n - 1, -2.0);
gsl_spmatrix_set(A, n - 1, n - 2, 1.0);
/* scale by h^2 */
gsl_spmatrix_scale(A, 1.0 / (h * h));
/* construct right hand side vector */
for (i = 0; i < n; ++i)
{
double xi = (i + 1) * h;
double fi = -M_PI * M_PI * sin(M_PI * xi);
gsl_vector_set(f, i, fi);
}
/* convert to compressed column format */
C = gsl_spmatrix_ccs(A);
/* now solve the system with the GMRES iterative solver */
{
const double tol = 1.0e-6; /* solution relative tolerance */
const size_t max_iter = 10; /* maximum iterations */
const gsl_splinalg_itersolve_type *T = gsl_splinalg_itersolve_gmres;
gsl_splinalg_itersolve *work =
gsl_splinalg_itersolve_alloc(T, n, 0);
size_t iter = 0;
double residual;
int status;
/* initial guess u = 0 */
gsl_vector_set_zero(u);
/* solve the system A u = f */
do
{
status = gsl_splinalg_itersolve_iterate(C, f, tol, u, work);
/* print out residual norm ||A*u - f|| */
residual = gsl_splinalg_itersolve_normr(work);
fprintf(stderr, "iter %zu residual = %.12e\n", iter, residual);
if (status == GSL_SUCCESS)
fprintf(stderr, "Converged\n");
}
while (status == GSL_CONTINUE && ++iter < max_iter);
/* output solution */
for (i = 0; i < n; ++i)
{
double xi = (i + 1) * h;
double u_exact = sin(M_PI * xi);
double u_gsl = gsl_vector_get(u, i);
printf("%f %.12e %.12e\n", xi, u_gsl, u_exact);
}
gsl_splinalg_itersolve_free(work);
}
gsl_spmatrix_free(A);
gsl_spmatrix_free(C);
gsl_vector_free(f);
gsl_vector_free(u);
return 0;
} /* main() */
Next: Sparse Linear Algebra References and Further Reading, Previous: Sparse Iterative Solvers, Up: Sparse Linear Algebra [Index]
Next: Cholesky Decomposition, Previous: Complete Orthogonal Decomposition, Up: Linear Algebra [Index]
A general rectangular M-by-N matrix A has a singular value decomposition (SVD) into the product of an M-by-N orthogonal matrix U, an N-by-N diagonal matrix of singular values S and the transpose of an N-by-N orthogonal square matrix V,
A = U S V^T
The singular values \sigma_i = S_{ii} are all non-negative and are generally chosen to form a non-increasing sequence \sigma_1 >= \sigma_2 >= ... >= \sigma_N >= 0.
The singular value decomposition of a matrix has many practical uses. The condition number of the matrix is given by the ratio of the largest singular value to the smallest singular value. The presence of a zero singular value indicates that the matrix is singular. The number of non-zero singular values indicates the rank of the matrix. In practice singular value decomposition of a rank-deficient matrix will not produce exact zeroes for singular values, due to finite numerical precision. Small singular values should be edited by choosing a suitable tolerance.
For a rank-deficient matrix, the null space of A is given by the columns of V corresponding to the zero singular values. Similarly, the range of A is given by columns of U corresponding to the non-zero singular values.
Note that the routines here compute the “thin” version of the SVD with U as M-by-N orthogonal matrix. This allows in-place computation and is the most commonly-used form in practice. Mathematically, the “full” SVD is defined with U as an M-by-M orthogonal matrix and S as an M-by-N diagonal matrix (with additional rows of zeros).
This function factorizes the M-by-N matrix A into the singular value decomposition A = U S V^T for M >= N. On output the matrix A is replaced by U. The diagonal elements of the singular value matrix S are stored in the vector S. The singular values are non-negative and form a non-increasing sequence from S_1 to S_N. The matrix V contains the elements of V in untransposed form. To form the product U S V^T it is necessary to take the transpose of V. A workspace of length N is required in work.
This routine uses the Golub-Reinsch SVD algorithm.
This function computes the SVD using the modified Golub-Reinsch algorithm, which is faster for M>>N. It requires the vector work of length N and the N-by-N matrix X as additional working space.
This function computes the SVD of the M-by-N matrix A using one-sided Jacobi orthogonalization for M >= N. The Jacobi method can compute singular values to higher relative accuracy than Golub-Reinsch algorithms (see references for details).
This function solves the system A x = b using the singular value
decomposition (U, S, V) of A which must
have been computed previously with gsl_linalg_SV_decomp.
Only non-zero singular values are used in computing the solution. The parts of the solution corresponding to singular values of zero are ignored. Other singular values can be edited out by setting them to zero before calling this function.
In the over-determined case where A has more rows than columns the system is solved in the least squares sense, returning the solution x which minimizes ||A x - b||_2.
This function computes the statistical leverage values h_i of a matrix A
using its singular value decomposition (U, S, V) previously computed
with gsl_linalg_SV_decomp. h_i are the diagonal values of the matrix
A (A^T A)^{-1} A^T and depend only on the matrix U which is the input to
this function.
Next: Cholesky Decomposition, Previous: Complete Orthogonal Decomposition, Up: Linear Algebra [Index]
Next: Error Handlers, Previous: Error Reporting, Up: Error Handling [Index]
The error code numbers returned by library functions are defined in
the file gsl_errno.h. They all have the prefix GSL_ and
expand to non-zero constant integer values. Error codes above 1024 are
reserved for applications, and are not used by the library. Many of
the error codes use the same base name as the corresponding error code
in the C library. Here are some of the most common error codes,
Domain error; used by mathematical functions when an argument value does not fall into the domain over which the function is defined (like EDOM in the C library)
Range error; used by mathematical functions when the result value is not representable because of overflow or underflow (like ERANGE in the C library)
No memory available. The system cannot allocate more virtual memory
because its capacity is full (like ENOMEM in the C library). This error
is reported when a GSL routine encounters problems when trying to
allocate memory with malloc.
Invalid argument. This is used to indicate various kinds of problems with passing the wrong argument to a library function (like EINVAL in the C library).
The error codes can be converted into an error message using the
function gsl_strerror.
This function returns a pointer to a string describing the error code gsl_errno. For example,
printf ("error: %s\n", gsl_strerror (status));
would print an error message like error: output range error for a
status value of GSL_ERANGE.
Next: Error Handlers, Previous: Error Reporting, Up: Error Handling [Index]
Next: Root Bracketing Algorithms, Previous: Root Finding Iteration, Up: One dimensional Root-Finding [Index]
A root finding procedure should stop when one of the following conditions is true:
The handling of these conditions is under user control. The functions below allow the user to test the precision of the current result in several standard ways.
This function tests for the convergence of the interval [x_lower,
x_upper] with absolute error epsabs and relative error
epsrel. The test returns GSL_SUCCESS if the following
condition is achieved,
|a - b| < epsabs + epsrel min(|a|,|b|)
when the interval x = [a,b] does not include the origin. If the interval includes the origin then \min(|a|,|b|) is replaced by zero (which is the minimum value of |x| over the interval). This ensures that the relative error is accurately estimated for roots close to the origin.
This condition on the interval also implies that any estimate of the root r in the interval satisfies the same condition with respect to the true root r^*,
|r - r^*| < epsabs + epsrel r^*
assuming that the true root r^* is contained within the interval.
This function tests for the convergence of the sequence …, x0,
x1 with absolute error epsabs and relative error
epsrel. The test returns GSL_SUCCESS if the following
condition is achieved,
|x_1 - x_0| < epsabs + epsrel |x_1|
and returns GSL_CONTINUE otherwise.
This function tests the residual value f against the absolute
error bound epsabs. The test returns GSL_SUCCESS if the
following condition is achieved,
|f| < epsabs
and returns GSL_CONTINUE otherwise. This criterion is suitable
for situations where the precise location of the root, x, is
unimportant provided a value can be found where the residual,
|f(x)|, is small enough.
Next: Root Bracketing Algorithms, Previous: Root Finding Iteration, Up: One dimensional Root-Finding [Index]
Next: The Bernoulli Distribution, Previous: General Discrete Distributions, Up: Random Number Distributions [Index]
This function returns a random integer from the Poisson distribution with mean mu. The probability distribution for Poisson variates is,
p(k) = {\mu^k \over k!} \exp(-\mu)
for k >= 0.
This function computes the probability p(k) of obtaining k from a Poisson distribution with mean mu, using the formula given above.
These functions compute the cumulative distribution functions P(k), Q(k) for the Poisson distribution with parameter mu.
Next: Nonlinear Least-Squares Testing for Convergence, Previous: Nonlinear Least-Squares Function Definition, Up: Nonlinear Least-Squares Fitting [Index]
The following functions drive the iteration of each algorithm. Each function performs one iteration of the trust region method and updates the state of the solver.
These functions perform a single iteration of the solver w. If the iteration encounters an unexpected problem then an error code will be returned. The solver workspace maintains a current estimate of the best-fit parameters at all times.
The solver workspace w contains the following entries, which can be used to track the progress of the solution:
gsl_vector * xThe current position, length p.
gsl_vector * fThe function residual vector at the current position f(x), length n.
gsl_matrix * JThe Jacobian matrix at the current position J(x), size
n-by-p (only for gsl_multifit_nlinear interface).
gsl_vector * dxThe difference between the current position and the previous position, i.e. the last step \delta, taken as a vector, length p.
These quantities can be accessed with the following functions,
These functions return the current position x (i.e. best-fit parameters) of the solver w.
These functions return the current residual vector f(x) of the solver w. For weighted systems, the residual vector includes the weighting factor \sqrt{W}.
This function returns a pointer to the n-by-p Jacobian matrix for the
current iteration of the solver w. This function is available only for the
gsl_multifit_nlinear interface.
These functions return the number of iterations performed so far.
The iteration counter is updated on each call to the
_iterate functions above, and reset to 0 in the
_init functions.
This function estimates the reciprocal condition number of the Jacobian matrix at the current position x and stores it in rcond. The computed value is only an estimate to give the user a guideline as to the conditioning of their particular problem. Its calculation is based on which factorization method is used (Cholesky, QR, or SVD).
Next: Nonlinear Least-Squares Testing for Convergence, Previous: Nonlinear Least-Squares Function Definition, Up: Nonlinear Least-Squares Fitting [Index]
Next: Roots of Polynomials References and Further Reading, Previous: General Polynomial Equations, Up: Polynomials [Index]
To demonstrate the use of the general polynomial solver we will take the polynomial P(x) = x^5 - 1 which has the following roots,
1, e^{2\pi i /5}, e^{4\pi i /5}, e^{6\pi i /5}, e^{8\pi i /5}
The following program will find these roots.
#include <stdio.h>
#include <gsl/gsl_poly.h>
int
main (void)
{
int i;
/* coefficients of P(x) = -1 + x^5 */
double a[6] = { -1, 0, 0, 0, 0, 1 };
double z[10];
gsl_poly_complex_workspace * w
= gsl_poly_complex_workspace_alloc (6);
gsl_poly_complex_solve (a, 6, w, z);
gsl_poly_complex_workspace_free (w);
for (i = 0; i < 5; i++)
{
printf ("z%d = %+.18f %+.18f\n",
i, z[2*i], z[2*i+1]);
}
return 0;
}
The output of the program is,
$ ./a.out
z0 = -0.809016994374947673 +0.587785252292473359 z1 = -0.809016994374947673 -0.587785252292473359 z2 = +0.309016994374947507 +0.951056516295152976 z3 = +0.309016994374947507 -0.951056516295152976 z4 = +0.999999999999999889 +0.000000000000000000
which agrees with the analytic result, z_n = \exp(2 \pi n i/5).
gsl-ref-html-2.3/Long-double.html 0000664 0001750 0001750 00000012160 13055414552 014774 0 ustar edd eddNext: Portability functions, Previous: Inline functions, Up: Using the library [Index]
In general, the algorithms in the library are written for double
precision only. The long double type is not supported for
actual computation.
One reason for this choice is that the precision of long double
is platform dependent. The IEEE standard only specifies the minimum
precision of extended precision numbers, while the precision of
double is the same on all platforms.
However, it is sometimes necessary to interact with external data in long-double format, so the vector and matrix datatypes include long-double versions.
It should be noted that in some system libraries the stdio.h
formatted input/output functions printf and scanf are
not implemented correctly for long double. Undefined or
incorrect results are avoided by testing these functions during the
configure stage of library compilation and eliminating certain
GSL functions which depend on them if necessary. The corresponding
line in the configure output looks like this,
checking whether printf works with long double... no
Consequently when long double formatted input/output does not
work on a given system it should be impossible to link a program which
uses GSL functions dependent on this.
If it is necessary to work on a system which does not support formatted
long double input/output then the options are to use binary
formats or to convert long double results into double for
reading and writing.
Next: Viscosity, Previous: Thermal Energy and Power, Up: Physical Constants [Index]
GSL_CONST_MKSA_BARThe pressure of 1 bar.
GSL_CONST_MKSA_STD_ATMOSPHEREThe pressure of 1 standard atmosphere.
GSL_CONST_MKSA_TORRThe pressure of 1 torr.
GSL_CONST_MKSA_METER_OF_MERCURYThe pressure of 1 meter of mercury.
GSL_CONST_MKSA_INCH_OF_MERCURYThe pressure of 1 inch of mercury.
GSL_CONST_MKSA_INCH_OF_WATERThe pressure of 1 inch of water.
GSL_CONST_MKSA_PSIThe pressure of 1 pound per square inch.
Next: Accessing permutation elements, Previous: The Permutation struct, Up: Permutations [Index]
This function allocates memory for a new permutation of size n.
The permutation is not initialized and its elements are undefined. Use
the function gsl_permutation_calloc if you want to create a
permutation which is initialized to the identity. A null pointer is
returned if insufficient memory is available to create the permutation.
This function allocates memory for a new permutation of size n and initializes it to the identity. A null pointer is returned if insufficient memory is available to create the permutation.
This function initializes the permutation p to the identity, i.e. (0,1,2,…,n-1).
This function frees all the memory used by the permutation p.
This function copies the elements of the permutation src into the permutation dest. The two permutations must have the same size.
Previous: Reading and writing multisets, Up: Multisets [Index]
The example program below prints all multisets elements containing the values {0,1,2,3} ordered by size. Multiset elements of the same size are ordered lexicographically.
#include <stdio.h>
#include <gsl/gsl_multiset.h>
int
main (void)
{
gsl_multiset * c;
size_t i;
printf ("All multisets of {0,1,2,3} by size:\n") ;
for (i = 0; i <= 4; i++)
{
c = gsl_multiset_calloc (4, i);
do
{
printf ("{");
gsl_multiset_fprintf (stdout, c, " %u");
printf (" }\n");
}
while (gsl_multiset_next (c) == GSL_SUCCESS);
gsl_multiset_free (c);
}
return 0;
}
Here is the output from the program,
$ ./a.out
All multisets of {0,1,2,3} by size:
{ }
{ 0 }
{ 1 }
{ 2 }
{ 3 }
{ 0 0 }
{ 0 1 }
{ 0 2 }
{ 0 3 }
{ 1 1 }
{ 1 2 }
{ 1 3 }
{ 2 2 }
{ 2 3 }
{ 3 3 }
{ 0 0 0 }
{ 0 0 1 }
{ 0 0 2 }
{ 0 0 3 }
{ 0 1 1 }
{ 0 1 2 }
{ 0 1 3 }
{ 0 2 2 }
{ 0 2 3 }
{ 0 3 3 }
{ 1 1 1 }
{ 1 1 2 }
{ 1 1 3 }
{ 1 2 2 }
{ 1 2 3 }
{ 1 3 3 }
{ 2 2 2 }
{ 2 2 3 }
{ 2 3 3 }
{ 3 3 3 }
{ 0 0 0 0 }
{ 0 0 0 1 }
{ 0 0 0 2 }
{ 0 0 0 3 }
{ 0 0 1 1 }
{ 0 0 1 2 }
{ 0 0 1 3 }
{ 0 0 2 2 }
{ 0 0 2 3 }
{ 0 0 3 3 }
{ 0 1 1 1 }
{ 0 1 1 2 }
{ 0 1 1 3 }
{ 0 1 2 2 }
{ 0 1 2 3 }
{ 0 1 3 3 }
{ 0 2 2 2 }
{ 0 2 2 3 }
{ 0 2 3 3 }
{ 0 3 3 3 }
{ 1 1 1 1 }
{ 1 1 1 2 }
{ 1 1 1 3 }
{ 1 1 2 2 }
{ 1 1 2 3 }
{ 1 1 3 3 }
{ 1 2 2 2 }
{ 1 2 2 3 }
{ 1 2 3 3 }
{ 1 3 3 3 }
{ 2 2 2 2 }
{ 2 2 2 3 }
{ 2 2 3 3 }
{ 2 3 3 3 }
{ 3 3 3 3 }
All 70 multisets are generated and sorted lexicographically.
Previous: Reading and writing multisets, Up: Multisets [Index]
Next: Bidiagonalization, Previous: Hessenberg Decomposition of Real Matrices, Up: Linear Algebra [Index]
A general real matrix pair (A, B) can be decomposed by orthogonal similarity transformations into the form
A = U H V^T B = U R V^T
where U and V are orthogonal, H is an upper Hessenberg matrix, and R is upper triangular. The Hessenberg-Triangular reduction is the first step in the generalized Schur decomposition for the generalized eigenvalue problem.
This function computes the Hessenberg-Triangular decomposition of the matrix pair (A, B). On output, H is stored in A, and R is stored in B. If U and V are provided (they may be null), the similarity transformations are stored in them. Additional workspace of length N is needed in work.
Next: Combinations, Previous: Vectors and Matrices, Up: Top [Index]
This chapter describes functions for creating and manipulating permutations. A permutation p is represented by an array of n integers in the range 0 to n-1, where each value p_i occurs once and only once. The application of a permutation p to a vector v yields a new vector v' where v'_i = v_{p_i}. For example, the array (0,1,3,2) represents a permutation which exchanges the last two elements of a four element vector. The corresponding identity permutation is (0,1,2,3).
Note that the permutations produced by the linear algebra routines correspond to the exchange of matrix columns, and so should be considered as applying to row-vectors in the form v' = v P rather than column-vectors, when permuting the elements of a vector.
The functions described in this chapter are defined in the header file gsl_permutation.h.
This function allocates a workspace for computing running statistics. The size of the workspace is O(1).
This function frees the memory associated with the workspace w.
This function resets the workspace w to its initial state, so it can begin working on a new set of data.
Each algorithm computes an approximation to a definite integral of the form,
I = \int_a^b f(x) w(x) dx
where w(x) is a weight function (for general integrands w(x)=1). The user provides absolute and relative error bounds (epsabs, epsrel) which specify the following accuracy requirement,
|RESULT - I| <= max(epsabs, epsrel |I|)
where RESULT is the numerical approximation obtained by the algorithm. The algorithms attempt to estimate the absolute error ABSERR = |RESULT - I| in such a way that the following inequality holds,
|RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)
In short, the routines return the first approximation which has an absolute error smaller than epsabs or a relative error smaller than epsrel.
Note that this is an either-or constraint, not simultaneous. To compute to a specified absolute error, set epsrel to zero. To compute to a specified relative error, set epsabs to zero. The routines will fail to converge if the error bounds are too stringent, but always return the best approximation obtained up to that stage.
The algorithms in QUADPACK use a naming convention based on the following letters,
Q- quadrature routineN- non-adaptive integratorA- adaptive integratorG- general integrand (user-defined)W- weight function with integrandS- singularities can be more readily integratedP- points of special difficulty can be suppliedI- infinite range of integrationO- oscillatory weight function, cos or sinF- Fourier integralC- Cauchy principal value
The algorithms are built on pairs of quadrature rules, a higher order rule and a lower order rule. The higher order rule is used to compute the best approximation to an integral over a small range. The difference between the results of the higher order rule and the lower order rule gives an estimate of the error in the approximation.
| • Integrands without weight functions: | ||
| • Integrands with weight functions: | ||
| • Integrands with singular weight functions: |
Next: Sparse Matrices Examples, Previous: Sparse Matrices Compressed Format, Up: Sparse Matrices [Index]
The gsl_spmatrix structure can be converted into the dense gsl_matrix
format and vice versa with the following routines.
This function converts the dense matrix A into sparse triplet format and stores the result in S.
This function converts the sparse matrix S into a dense matrix and stores the result in A. S must be in triplet format.
Next: Permutation allocation, Up: Permutations [Index]
A permutation is defined by a structure containing two components, the size
of the permutation and a pointer to the permutation array. The elements
of the permutation array are all of type size_t. The
gsl_permutation structure looks like this,
typedef struct
{
size_t size;
size_t * data;
} gsl_permutation;
Next: Sorting References and Further Reading, Previous: Computing the rank, Up: Sorting [Index]
The following example shows how to use the permutation p to print the elements of the vector v in ascending order,
gsl_sort_vector_index (p, v);
for (i = 0; i < v->size; i++)
{
double vpi = gsl_vector_get (v, p->data[i]);
printf ("order = %d, value = %g\n", i, vpi);
}
The next example uses the function gsl_sort_smallest to select
the 5 smallest numbers from 100000 uniform random variates stored in an
array,
#include <gsl/gsl_rng.h>
#include <gsl/gsl_sort_double.h>
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
size_t i, k = 5, N = 100000;
double * x = malloc (N * sizeof(double));
double * small = malloc (k * sizeof(double));
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
for (i = 0; i < N; i++)
{
x[i] = gsl_rng_uniform(r);
}
gsl_sort_smallest (small, k, x, 1, N);
printf ("%zu smallest values from %zu\n", k, N);
for (i = 0; i < k; i++)
{
printf ("%zu: %.18f\n", i, small[i]);
}
free (x);
free (small);
gsl_rng_free (r);
return 0;
}
The output lists the 5 smallest values, in ascending order,
$ ./a.out
5 smallest values from 100000 0: 0.000003489200025797 1: 0.000008199829608202 2: 0.000008953968062997 3: 0.000010712770745158 4: 0.000033531803637743
Next: Finding maximum and minimum elements of matrices, Previous: Exchanging rows and columns, Up: Matrices [Index]
The following operations are defined for real and complex matrices.
This function adds the elements of matrix b to the elements of matrix a. The result a(i,j) \leftarrow a(i,j) + b(i,j) is stored in a and b remains unchanged. The two matrices must have the same dimensions.
This function subtracts the elements of matrix b from the elements of matrix a. The result a(i,j) \leftarrow a(i,j) - b(i,j) is stored in a and b remains unchanged. The two matrices must have the same dimensions.
This function multiplies the elements of matrix a by the elements of matrix b. The result a(i,j) \leftarrow a(i,j) * b(i,j) is stored in a and b remains unchanged. The two matrices must have the same dimensions.
This function divides the elements of matrix a by the elements of matrix b. The result a(i,j) \leftarrow a(i,j) / b(i,j) is stored in a and b remains unchanged. The two matrices must have the same dimensions.
This function multiplies the elements of matrix a by the constant factor x. The result a(i,j) \leftarrow x a(i,j) is stored in a.
This function adds the constant value x to the elements of the matrix a. The result a(i,j) \leftarrow a(i,j) + x is stored in a.
Next: Finding maximum and minimum elements of matrices, Previous: Exchanging rows and columns, Up: Matrices [Index]
Next: Two dimensional histograms, Previous: The histogram probability distribution struct, Up: Histograms [Index]
The following program shows how to make a simple histogram of a column
of numerical data supplied on stdin. The program takes three
arguments, specifying the upper and lower bounds of the histogram and
the number of bins. It then reads numbers from stdin, one line at
a time, and adds them to the histogram. When there is no more data to
read it prints out the accumulated histogram using
gsl_histogram_fprintf.
#include <stdio.h>
#include <stdlib.h>
#include <gsl/gsl_histogram.h>
int
main (int argc, char **argv)
{
double a, b;
size_t n;
if (argc != 4)
{
printf ("Usage: gsl-histogram xmin xmax n\n"
"Computes a histogram of the data "
"on stdin using n bins from xmin "
"to xmax\n");
exit (0);
}
a = atof (argv[1]);
b = atof (argv[2]);
n = atoi (argv[3]);
{
double x;
gsl_histogram * h = gsl_histogram_alloc (n);
gsl_histogram_set_ranges_uniform (h, a, b);
while (fscanf (stdin, "%lg", &x) == 1)
{
gsl_histogram_increment (h, x);
}
gsl_histogram_fprintf (stdout, h, "%g", "%g");
gsl_histogram_free (h);
}
exit (0);
}
Here is an example of the program in use. We generate 10000 random samples from a Cauchy distribution with a width of 30 and histogram them over the range -100 to 100, using 200 bins.
$ gsl-randist 0 10000 cauchy 30 | gsl-histogram -100 100 200 > histogram.dat
A plot of the resulting histogram shows the familiar shape of the Cauchy distribution and the fluctuations caused by the finite sample size.
$ awk '{print $1, $3 ; print $2, $3}' histogram.dat
| graph -T X
Next: Irregular Modified Spherical Bessel Functions, Previous: Irregular Spherical Bessel Functions, Up: Bessel Functions [Index]
The regular modified spherical Bessel functions i_l(x) are related to the modified Bessel functions of fractional order, i_l(x) = \sqrt{\pi/(2x)} I_{l+1/2}(x)
These routines compute the scaled regular modified spherical Bessel function of zeroth order, \exp(-|x|) i_0(x).
These routines compute the scaled regular modified spherical Bessel function of first order, \exp(-|x|) i_1(x).
These routines compute the scaled regular modified spherical Bessel function of second order, \exp(-|x|) i_2(x)
These routines compute the scaled regular modified spherical Bessel function of order l, \exp(-|x|) i_l(x)
This routine computes the values of the scaled regular modified spherical Bessel functions \exp(-|x|) i_l(x) for l from 0 to lmax inclusive for lmax >= 0, storing the results in the array result_array. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
Next: References and Further Reading for Multidimensional Root Finding, Previous: Algorithms without Derivatives, Up: Multidimensional Root-Finding [Index]
The multidimensional solvers are used in a similar way to the
one-dimensional root finding algorithms. This first example
demonstrates the hybrids scaled-hybrid algorithm, which does not
require derivatives. The program solves the Rosenbrock system of equations,
f_1 (x, y) = a (1 - x) f_2 (x, y) = b (y - x^2)
with a = 1, b = 10. The solution of this system lies at (x,y) = (1,1) in a narrow valley.
The first stage of the program is to define the system of equations,
#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_multiroots.h>
struct rparams
{
double a;
double b;
};
int
rosenbrock_f (const gsl_vector * x, void *params,
gsl_vector * f)
{
double a = ((struct rparams *) params)->a;
double b = ((struct rparams *) params)->b;
const double x0 = gsl_vector_get (x, 0);
const double x1 = gsl_vector_get (x, 1);
const double y0 = a * (1 - x0);
const double y1 = b * (x1 - x0 * x0);
gsl_vector_set (f, 0, y0);
gsl_vector_set (f, 1, y1);
return GSL_SUCCESS;
}
The main program begins by creating the function object f, with
the arguments (x,y) and parameters (a,b). The solver
s is initialized to use this function, with the hybrids
method.
int
main (void)
{
const gsl_multiroot_fsolver_type *T;
gsl_multiroot_fsolver *s;
int status;
size_t i, iter = 0;
const size_t n = 2;
struct rparams p = {1.0, 10.0};
gsl_multiroot_function f = {&rosenbrock_f, n, &p};
double x_init[2] = {-10.0, -5.0};
gsl_vector *x = gsl_vector_alloc (n);
gsl_vector_set (x, 0, x_init[0]);
gsl_vector_set (x, 1, x_init[1]);
T = gsl_multiroot_fsolver_hybrids;
s = gsl_multiroot_fsolver_alloc (T, 2);
gsl_multiroot_fsolver_set (s, &f, x);
print_state (iter, s);
do
{
iter++;
status = gsl_multiroot_fsolver_iterate (s);
print_state (iter, s);
if (status) /* check if solver is stuck */
break;
status =
gsl_multiroot_test_residual (s->f, 1e-7);
}
while (status == GSL_CONTINUE && iter < 1000);
printf ("status = %s\n", gsl_strerror (status));
gsl_multiroot_fsolver_free (s);
gsl_vector_free (x);
return 0;
}
Note that it is important to check the return status of each solver step, in case the algorithm becomes stuck. If an error condition is detected, indicating that the algorithm cannot proceed, then the error can be reported to the user, a new starting point chosen or a different algorithm used.
The intermediate state of the solution is displayed by the following
function. The solver state contains the vector s->x which is the
current position, and the vector s->f with corresponding function
values.
int
print_state (size_t iter, gsl_multiroot_fsolver * s)
{
printf ("iter = %3u x = % .3f % .3f "
"f(x) = % .3e % .3e\n",
iter,
gsl_vector_get (s->x, 0),
gsl_vector_get (s->x, 1),
gsl_vector_get (s->f, 0),
gsl_vector_get (s->f, 1));
}
Here are the results of running the program. The algorithm is started at (-10,-5) far from the solution. Since the solution is hidden in a narrow valley the earliest steps follow the gradient of the function downhill, in an attempt to reduce the large value of the residual. Once the root has been approximately located, on iteration 8, the Newton behavior takes over and convergence is very rapid.
iter = 0 x = -10.000 -5.000 f(x) = 1.100e+01 -1.050e+03 iter = 1 x = -10.000 -5.000 f(x) = 1.100e+01 -1.050e+03 iter = 2 x = -3.976 24.827 f(x) = 4.976e+00 9.020e+01 iter = 3 x = -3.976 24.827 f(x) = 4.976e+00 9.020e+01 iter = 4 x = -3.976 24.827 f(x) = 4.976e+00 9.020e+01 iter = 5 x = -1.274 -5.680 f(x) = 2.274e+00 -7.302e+01 iter = 6 x = -1.274 -5.680 f(x) = 2.274e+00 -7.302e+01 iter = 7 x = 0.249 0.298 f(x) = 7.511e-01 2.359e+00 iter = 8 x = 0.249 0.298 f(x) = 7.511e-01 2.359e+00 iter = 9 x = 1.000 0.878 f(x) = 1.268e-10 -1.218e+00 iter = 10 x = 1.000 0.989 f(x) = 1.124e-11 -1.080e-01 iter = 11 x = 1.000 1.000 f(x) = 0.000e+00 0.000e+00 status = success
Note that the algorithm does not update the location on every iteration. Some iterations are used to adjust the trust-region parameter, after trying a step which was found to be divergent, or to recompute the Jacobian, when poor convergence behavior is detected.
The next example program adds derivative information, in order to
accelerate the solution. There are two derivative functions
rosenbrock_df and rosenbrock_fdf. The latter computes both
the function and its derivative simultaneously. This allows the
optimization of any common terms. For simplicity we substitute calls to
the separate f and df functions at this point in the code
below.
int
rosenbrock_df (const gsl_vector * x, void *params,
gsl_matrix * J)
{
const double a = ((struct rparams *) params)->a;
const double b = ((struct rparams *) params)->b;
const double x0 = gsl_vector_get (x, 0);
const double df00 = -a;
const double df01 = 0;
const double df10 = -2 * b * x0;
const double df11 = b;
gsl_matrix_set (J, 0, 0, df00);
gsl_matrix_set (J, 0, 1, df01);
gsl_matrix_set (J, 1, 0, df10);
gsl_matrix_set (J, 1, 1, df11);
return GSL_SUCCESS;
}
int
rosenbrock_fdf (const gsl_vector * x, void *params,
gsl_vector * f, gsl_matrix * J)
{
rosenbrock_f (x, params, f);
rosenbrock_df (x, params, J);
return GSL_SUCCESS;
}
The main program now makes calls to the corresponding fdfsolver
versions of the functions,
int
main (void)
{
const gsl_multiroot_fdfsolver_type *T;
gsl_multiroot_fdfsolver *s;
int status;
size_t i, iter = 0;
const size_t n = 2;
struct rparams p = {1.0, 10.0};
gsl_multiroot_function_fdf f = {&rosenbrock_f,
&rosenbrock_df,
&rosenbrock_fdf,
n, &p};
double x_init[2] = {-10.0, -5.0};
gsl_vector *x = gsl_vector_alloc (n);
gsl_vector_set (x, 0, x_init[0]);
gsl_vector_set (x, 1, x_init[1]);
T = gsl_multiroot_fdfsolver_gnewton;
s = gsl_multiroot_fdfsolver_alloc (T, n);
gsl_multiroot_fdfsolver_set (s, &f, x);
print_state (iter, s);
do
{
iter++;
status = gsl_multiroot_fdfsolver_iterate (s);
print_state (iter, s);
if (status)
break;
status = gsl_multiroot_test_residual (s->f, 1e-7);
}
while (status == GSL_CONTINUE && iter < 1000);
printf ("status = %s\n", gsl_strerror (status));
gsl_multiroot_fdfsolver_free (s);
gsl_vector_free (x);
return 0;
}
The addition of derivative information to the hybrids solver does
not make any significant difference to its behavior, since it able to
approximate the Jacobian numerically with sufficient accuracy. To
illustrate the behavior of a different derivative solver we switch to
gnewton. This is a traditional Newton solver with the constraint
that it scales back its step if the full step would lead “uphill”. Here
is the output for the gnewton algorithm,
iter = 0 x = -10.000 -5.000 f(x) = 1.100e+01 -1.050e+03 iter = 1 x = -4.231 -65.317 f(x) = 5.231e+00 -8.321e+02 iter = 2 x = 1.000 -26.358 f(x) = -8.882e-16 -2.736e+02 iter = 3 x = 1.000 1.000 f(x) = -2.220e-16 -4.441e-15 status = success
The convergence is much more rapid, but takes a wide excursion out to the point (-4.23,-65.3). This could cause the algorithm to go astray in a realistic application. The hybrid algorithm follows the downhill path to the solution more reliably.
Next: References and Further Reading for Multidimensional Root Finding, Previous: Algorithms without Derivatives, Up: Multidimensional Root-Finding [Index]
Next: No Warranty, Previous: GSL is Free Software, Up: Introduction [Index]
The source code for the library can be obtained in different ways, by copying it from a friend, purchasing it on CDROM or downloading it from the internet. A list of public ftp servers which carry the source code can be found on the GNU website,
The preferred platform for the library is a GNU system, which allows it to take advantage of additional features in the GNU C compiler and GNU C library. However, the library is fully portable and should compile on most systems with a C compiler.
Announcements of new releases, updates and other relevant events are
made on the info-gsl@gnu.org mailing list. To subscribe to this
low-volume list, send an email of the following form:
To: info-gsl-request@gnu.org Subject: subscribe
You will receive a response asking you to reply in order to confirm your subscription.
gsl-ref-html-2.3/Multimin-Stopping-Criteria.html 0000664 0001750 0001750 00000012666 13055414473 020001 0 ustar edd eddNext: Multimin Algorithms with Derivatives, Previous: Multimin Iteration, Up: Multidimensional Minimization [Index]
A minimization procedure should stop when one of the following conditions is true:
The handling of these conditions is under user control. The functions below allow the user to test the precision of the current result.
This function tests the norm of the gradient g against the
absolute tolerance epsabs. The gradient of a multidimensional
function goes to zero at a minimum. The test returns GSL_SUCCESS
if the following condition is achieved,
|g| < epsabs
and returns GSL_CONTINUE otherwise. A suitable choice of
epsabs can be made from the desired accuracy in the function for
small variations in x. The relationship between these quantities
is given by \delta f = g \delta x.
This function tests the minimizer specific characteristic
size (if applicable to the used minimizer) against absolute tolerance epsabs.
The test returns GSL_SUCCESS if the size is smaller than tolerance,
otherwise GSL_CONTINUE is returned.
Previous: Example ntuple programs, Up: N-tuples [Index]
Further information on the use of ntuples can be found in the documentation for the CERN packages PAW and HBOOK (available online).
gsl-ref-html-2.3/Nonlinear-Least_002dSquares-Covariance-Matrix.html 0000664 0001750 0001750 00000016346 13055414472 023236 0 ustar edd eddNext: Nonlinear Least-Squares Troubleshooting, Previous: Nonlinear Least-Squares High Level Driver, Up: Nonlinear Least-Squares Fitting [Index]
This function computes the covariance matrix of best-fit parameters using the Jacobian matrix J and stores it in covar. The parameter epsrel is used to remove linear-dependent columns when J is rank deficient.
The covariance matrix is given by,
covar = (J^T J)^{-1}
or in the weighted case,
covar = (J^T W J)^{-1}
and is computed using the factored form of the Jacobian (Cholesky, QR, or SVD). Any columns of R which satisfy
|R_{kk}| <= epsrel |R_{11}|
are considered linearly-dependent and are excluded from the covariance matrix (the corresponding rows and columns of the covariance matrix are set to zero).
If the minimisation uses the weighted least-squares function f_i = (Y(x, t_i) - y_i) / \sigma_i then the covariance matrix above gives the statistical error on the best-fit parameters resulting from the Gaussian errors \sigma_i on the underlying data y_i. This can be verified from the relation \delta f = J \delta c and the fact that the fluctuations in f from the data y_i are normalised by \sigma_i and so satisfy <\delta f \delta f^T> = I.
For an unweighted least-squares function f_i = (Y(x, t_i) - y_i) the covariance matrix above should be multiplied by the variance of the residuals about the best-fit \sigma^2 = \sum (y_i - Y(x,t_i))^2 / (n-p) to give the variance-covariance matrix \sigma^2 C. This estimates the statistical error on the best-fit parameters from the scatter of the underlying data.
For more information about covariance matrices see Fitting Overview.
Next: Nonlinear Least-Squares Troubleshooting, Previous: Nonlinear Least-Squares High Level Driver, Up: Nonlinear Least-Squares Fitting [Index]
Next: Reading and writing permutations, Previous: Permutation functions, Up: Permutations [Index]
This function applies the permutation p to the array data of size n with stride stride.
This function applies the inverse of the permutation p to the array data of size n with stride stride.
This function applies the permutation p to the elements of the vector v, considered as a row-vector acted on by a permutation matrix from the right, v' = v P. The j-th column of the permutation matrix P is given by the p_j-th column of the identity matrix. The permutation p and the vector v must have the same length.
This function applies the inverse of the permutation p to the elements of the vector v, considered as a row-vector acted on by an inverse permutation matrix from the right, v' = v P^T. Note that for permutation matrices the inverse is the same as the transpose. The j-th column of the permutation matrix P is given by the p_j-th column of the identity matrix. The permutation p and the vector v must have the same length.
This function applies the permutation p to the matrix A from the right, A' = A P. The j-th column of the permutation matrix P is given by the p_j-th column of the identity matrix. This effectively permutes the columns of A according to the permutation p, and so the number of columns of A must equal the size of the permutation p.
This function combines the two permutations pa and pb into a single permutation p, where p = pa * pb. The permutation p is equivalent to applying pb first and then pa.
Next: Reading and writing permutations, Previous: Permutation functions, Up: Permutations [Index]
Next: 2D Evaluation of Interpolating Functions, Previous: 2D Interpolation Grids, Up: Interpolation [Index]
The interpolation library provides the following 2D interpolation types:
Bilinear interpolation. This interpolation method does not require any additional memory.
This function returns the name of the interpolation type used by interp. For example,
printf ("interp uses '%s' interpolation.\n",
gsl_interp2d_name (interp));
would print something like,
interp uses 'bilinear' interpolation.
These functions return the minimum number of points required by the interpolation object interp or interpolation type T. For example, bicubic interpolation requires a minimum of 4 points.
Next: The Cauchy Distribution, Previous: The Laplace Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the exponential power distribution with scale parameter a and exponent b. The distribution is,
p(x) dx = {1 \over 2 a \Gamma(1+1/b)} \exp(-|x/a|^b) dx
for x >= 0. For b = 1 this reduces to the Laplace distribution. For b = 2 it has the same form as a Gaussian distribution, but with a = \sqrt{2} \sigma.
This function computes the probability density p(x) at x for an exponential power distribution with scale parameter a and exponent b, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) for the exponential power distribution with parameters a and b.
Next: Relative Exponential Functions, Up: Exponential Functions [Index]
These routines provide an exponential function \exp(x) using GSL semantics and error checking.
This function computes the exponential \exp(x) using the
gsl_sf_result_e10 type to return a result with extended range.
This function may be useful if the value of \exp(x) would
overflow the numeric range of double.
These routines exponentiate x and multiply by the factor y to return the product y \exp(x).
This function computes the product y \exp(x) using the
gsl_sf_result_e10 type to return a result with extended numeric
range.
Next: Example programs for blocks, Previous: Block allocation, Up: Blocks [Index]
The library provides functions for reading and writing blocks to a file as binary data or formatted text.
This function writes the elements of the block b to the stream
stream in binary format. The return value is 0 for success and
GSL_EFAILED if there was a problem writing to the file. Since the
data is written in the native binary format it may not be portable
between different architectures.
This function reads into the block b from the open stream
stream in binary format. The block b must be preallocated
with the correct length since the function uses the size of b to
determine how many bytes to read. The return value is 0 for success and
GSL_EFAILED if there was a problem reading from the file. The
data is assumed to have been written in the native binary format on the
same architecture.
This function writes the elements of the block b line-by-line to
the stream stream using the format specifier format, which
should be one of the %g, %e or %f formats for
floating point numbers and %d for integers. The function returns
0 for success and GSL_EFAILED if there was a problem writing to
the file.
This function reads formatted data from the stream stream into the
block b. The block b must be preallocated with the correct
length since the function uses the size of b to determine how many
numbers to read. The function returns 0 for success and
GSL_EFAILED if there was a problem reading from the file.
Next: Example programs for blocks, Previous: Block allocation, Up: Blocks [Index]
Next: Dawson Function, Previous: Coulomb Functions, Up: Special Functions [Index]
The Wigner 3-j, 6-j and 9-j symbols give the coupling coefficients for combined angular momentum vectors. Since the arguments of the standard coupling coefficient functions are integer or half-integer, the arguments of the following functions are, by convention, integers equal to twice the actual spin value. For information on the 3-j coefficients see Abramowitz & Stegun, Section 27.9. The functions described in this section are declared in the header file gsl_sf_coupling.h.
| • 3-j Symbols: | ||
| • 6-j Symbols: | ||
| • 9-j Symbols: |
Next: Nonlinear Least-Squares Tunable Parameters, Previous: Nonlinear Least-Squares TRS Overview, Up: Nonlinear Least-Squares Fitting [Index]
Weighted nonlinear least-squares fitting minimizes the function
\Phi(x) = (1/2) || f(x) ||_W^2
= (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2
where W = diag(w_1,w_2,...,w_n) is the weighting matrix,
and ||f||_W^2 = f^T W f.
The weights w_i are commonly defined as w_i = 1/\sigma_i^2,
where \sigma_i is the error in the ith measurement.
A simple change of variables \tilde{f} = W^{1 \over 2} f yields
\Phi(x) = {1 \over 2} ||\tilde{f}||^2, which is in the
same form as the unweighted case. The user can either perform this
transform directly on their function residuals and Jacobian, or use
the gsl_multifit_nlinear_winit interface which automatically
performs the correct scaling. To manually perform this transformation,
the residuals and Jacobian should be modified according to
f~_i = f_i / \sigma_i J~_ij = 1 / \sigma_i df_i/dx_j
For large systems, the user must perform their own weighting.
gsl-ref-html-2.3/Trigonometric-Functions-for-Complex-Arguments.html 0000664 0001750 0001750 00000012500 13055414522 023547 0 ustar edd eddNext: Hyperbolic Trigonometric Functions, Previous: Circular Trigonometric Functions, Up: Trigonometric Functions [Index]
This function computes the complex sine, \sin(z_r + i z_i) storing the real and imaginary parts in szr, szi.
This function computes the complex cosine, \cos(z_r + i z_i) storing the real and imaginary parts in czr, czi.
This function computes the logarithm of the complex sine, \log(\sin(z_r + i z_i)) storing the real and imaginary parts in lszr, lszi.
Next: Search Stopping Parameters for the multidimensional solver, Previous: Providing the multidimensional system of equations to solve, Up: Multidimensional Root-Finding [Index]
The following functions drive the iteration of each algorithm. Each function performs one iteration to update the state of any solver of the corresponding type. The same functions work for all solvers so that different methods can be substituted at runtime without modifications to the code.
These functions perform a single iteration of the solver s. If the iteration encounters an unexpected problem then an error code will be returned,
GSL_EBADFUNCthe iteration encountered a singular point where the function or its
derivative evaluated to Inf or NaN.
GSL_ENOPROGthe iteration is not making any progress, preventing the algorithm from continuing.
The solver maintains a current best estimate of the root s->x
and its function value s->f at all times. This information can
be accessed with the following auxiliary functions,
These functions return the current estimate of the root for the solver s, given by s->x.
These functions return the function value f(x) at the current
estimate of the root for the solver s, given by s->f.
These functions return the last step dx taken by the solver
s, given by s->dx.
Next: Search Stopping Parameters for the multidimensional solver, Previous: Providing the multidimensional system of equations to solve, Up: Multidimensional Root-Finding [Index]
Next: Small integer powers, Previous: Infinities and Not-a-number, Up: Mathematical Functions [Index]
The following routines provide portable implementations of functions
found in the BSD math library. When native versions are not available
the functions described here can be used instead. The substitution can
be made automatically if you use autoconf to compile your
application (see Portability functions).
This function computes the value of \log(1+x) in a way that is
accurate for small x. It provides an alternative to the BSD math
function log1p(x).
This function computes the value of \exp(x)-1 in a way that is
accurate for small x. It provides an alternative to the BSD math
function expm1(x).
This function computes the value of
\sqrt{x^2 + y^2} in a way that avoids overflow. It provides an
alternative to the BSD math function hypot(x,y).
This function computes the value of \sqrt{x^2 + y^2 + z^2} in a way that avoids overflow.
This function computes the value of \arccosh(x). It provides an
alternative to the standard math function acosh(x).
This function computes the value of \arcsinh(x). It provides an
alternative to the standard math function asinh(x).
This function computes the value of \arctanh(x). It provides an
alternative to the standard math function atanh(x).
This function computes the value of x * 2^e. It provides an
alternative to the standard math function ldexp(x,e).
This function splits the number x into its normalized fraction
f and exponent e, such that x = f * 2^e and
0.5 <= f < 1. The function returns f and stores the
exponent in e. If x is zero, both f and e
are set to zero. This function provides an alternative to the standard
math function frexp(x, e).
Next: Small integer powers, Previous: Infinities and Not-a-number, Up: Mathematical Functions [Index]
Next: Chebyshev Series Evaluation, Previous: Creation and Calculation of Chebyshev Series, Up: Chebyshev Approximations [Index]
The following functions provide information about an existing Chebyshev series.
This function returns the order of Chebyshev series cs.
These functions return the size of the Chebyshev coefficient array
c[] and a pointer to its location in memory for the Chebyshev
series cs.
Next: Reading and writing random number generator state, Previous: Random number environment variables, Up: Random Number Generation [Index]
The above methods do not expose the random number ‘state’ which changes from call to call. It is often useful to be able to save and restore the state. To permit these practices, a few somewhat more advanced functions are supplied. These include:
This function copies the random number generator src into the pre-existing generator dest, making dest into an exact copy of src. The two generators must be of the same type.
This function returns a pointer to a newly created generator which is an exact copy of the generator r.
Next: Nonlinear Least-Squares High Level Driver, Previous: Nonlinear Least-Squares Iteration, Up: Nonlinear Least-Squares Fitting [Index]
A minimization procedure should stop when one of the following conditions is true:
The handling of these conditions is under user control. The functions below allow the user to test the current estimate of the best-fit parameters in several standard ways.
These functions test for convergence of the minimization method using the following criteria:
|\delta_i| <= xtol (|x_i| + xtol)
for each 0 <= i < p. Each element of the step vector \delta
is tested individually in case the different parameters have widely
different scales. Adding xtol to |x_i| helps the test avoid
breaking down in situations where the true solution value x_i = 0.
If this test succeeds, info is set to 1 and the function
returns GSL_SUCCESS.
A general guideline for selecting the step tolerance is to choose xtol = 10^{-d} where d is the number of accurate decimal digits desired in the solution x. See Dennis and Schnabel for more information.
||g||_inf <= gtol
This expression tests whether the ratio
(\nabla \Phi)_i x_i / \Phi is small. Testing this scaled gradient
is a better than \nabla \Phi alone since it is a dimensionless
quantity and so independent of the scale of the problem. The
max arguments help ensure the test doesn’t break down in
regions where x_i or \Phi(x) are close to 0.
If this test succeeds, info is set to 2 and the function
returns GSL_SUCCESS.
A general guideline for choosing the gradient tolerance is to set
gtol = GSL_DBL_EPSILON^(1/3). See Dennis and Schnabel for
more information.
If none of the tests succeed, info is set to 0 and the
function returns GSL_CONTINUE, indicating further iterations
are required.
Next: Nonlinear Least-Squares High Level Driver, Previous: Nonlinear Least-Squares Iteration, Up: Nonlinear Least-Squares Fitting [Index]
Previous: Physical Constant Examples, Up: Physical Constants [Index]
The authoritative sources for physical constants are the 2006 CODATA recommended values, published in the article below. Further information on the values of physical constants is also available from the NIST website.
Next: Copying rows and columns, Previous: Creating row and column views, Up: Matrices [Index]
This function copies the elements of the matrix src into the matrix dest. The two matrices must have the same size.
This function exchanges the elements of the matrices m1 and m2 by copying. The two matrices must have the same size.
Next: Multi-parameter regression, Previous: Fitting Overview, Up: Least-Squares Fitting [Index]
The functions in this section are used to fit simple one or two parameter linear regression models. The functions are declared in the header file gsl_fit.h.
| • Linear regression with a constant term: | ||
| • Linear regression without a constant term: |
Next: Sparse Matrices Properties, Previous: Sparse Matrices Exchanging Rows and Columns, Up: Sparse Matrices [Index]
This function computes the sum c = a + b. The three matrices must have the same dimensions and be stored in a compressed format.
This function scales all elements of the matrix m by the constant factor x. The result m(i,j) \leftarrow x m(i,j) is stored in m.
Next: Absolute deviation, Up: Statistics [Index]
This function returns the arithmetic mean of data, a dataset of length n with stride stride. The arithmetic mean, or sample mean, is denoted by \Hat\mu and defined as,
\Hat\mu = (1/N) \sum x_i
where x_i are the elements of the dataset data. For samples drawn from a gaussian distribution the variance of \Hat\mu is \sigma^2 / N.
This function returns the estimated, or sample, variance of data, a dataset of length n with stride stride. The estimated variance is denoted by \Hat\sigma^2 and is defined by,
\Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2
where x_i are the elements of the dataset data. Note that the normalization factor of 1/(N-1) results from the derivation of \Hat\sigma^2 as an unbiased estimator of the population variance \sigma^2. For samples drawn from a Gaussian distribution the variance of \Hat\sigma^2 itself is 2 \sigma^4 / N.
This function computes the mean via a call to gsl_stats_mean. If
you have already computed the mean then you can pass it directly to
gsl_stats_variance_m.
This function returns the sample variance of data relative to the given value of mean. The function is computed with \Hat\mu replaced by the value of mean that you supply,
\Hat\sigma^2 = (1/(N-1)) \sum (x_i - mean)^2
The standard deviation is defined as the square root of the variance. These functions return the square root of the corresponding variance functions above.
These functions return the total sum of squares (TSS) of data about
the mean. For gsl_stats_tss_m the user-supplied value of
mean is used, and for gsl_stats_tss it is computed using
gsl_stats_mean.
TSS = \sum (x_i - mean)^2
This function computes an unbiased estimate of the variance of data when the population mean mean of the underlying distribution is known a priori. In this case the estimator for the variance uses the factor 1/N and the sample mean \Hat\mu is replaced by the known population mean \mu,
\Hat\sigma^2 = (1/N) \sum (x_i - \mu)^2
This function calculates the standard deviation of data for a fixed population mean mean. The result is the square root of the corresponding variance function.
Next: Absolute deviation, Up: Statistics [Index]
Next: 1D Interpolation Types, Previous: 1D Introduction to Interpolation, Up: Interpolation [Index]
The interpolation function for a given dataset is stored in a
gsl_interp object. These are created by the following functions.
This function returns a pointer to a newly allocated interpolation object of type T for size data-points.
This function initializes the interpolation object interp for the
data (xa,ya) where xa and ya are arrays of size
size. The interpolation object (gsl_interp) does not save
the data arrays xa and ya and only stores the static state
computed from the data. The xa data array is always assumed to be
strictly ordered, with increasing x values;
the behavior for other arrangements is not defined.
This function frees the interpolation object interp.
Next: Matrix operations, Previous: Copying rows and columns, Up: Matrices [Index]
The following functions can be used to exchange the rows and columns of a matrix.
This function exchanges the i-th and j-th rows of the matrix m in-place.
This function exchanges the i-th and j-th columns of the matrix m in-place.
This function exchanges the i-th row and j-th column of the matrix m in-place. The matrix must be square for this operation to be possible.
This function makes the matrix dest the transpose of the matrix src by copying the elements of src into dest. This function works for all matrices provided that the dimensions of the matrix dest match the transposed dimensions of the matrix src.
This function replaces the matrix m by its transpose by copying the elements of the matrix in-place. The matrix must be square for this operation to be possible.
Next: Accessing multiset elements, Previous: The Multiset struct, Up: Multisets [Index]
This function allocates memory for a new multiset with parameters n,
k. The multiset is not initialized and its elements are undefined. Use
the function gsl_multiset_calloc if you want to create a multiset which
is initialized to the lexicographically first multiset element. A null pointer
is returned if insufficient memory is available to create the multiset.
This function allocates memory for a new multiset with parameters n, k and initializes it to the lexicographically first multiset element. A null pointer is returned if insufficient memory is available to create the multiset.
This function initializes the multiset c to the lexicographically first multiset element, i.e. 0 repeated k times.
This function initializes the multiset c to the lexicographically last multiset element, i.e. n-1 repeated k times.
This function frees all the memory used by the multiset c.
This function copies the elements of the multiset src into the multiset dest. The two multisets must have the same size.
Next: Pressure, Previous: Mass and Weight, Up: Physical Constants [Index]
GSL_CONST_MKSA_CALORIEThe energy of 1 calorie.
GSL_CONST_MKSA_BTUThe energy of 1 British Thermal Unit, btu.
GSL_CONST_MKSA_THERMThe energy of 1 Therm.
GSL_CONST_MKSA_HORSEPOWERThe power of 1 horsepower.
Previous: Trivial example, Up: Examples with Simulated Annealing [Index]
The TSP (Traveling Salesman Problem) is the classic combinatorial optimization problem. I have provided a very simple version of it, based on the coordinates of twelve cities in the southwestern United States. This should maybe be called the Flying Salesman Problem, since I am using the great-circle distance between cities, rather than the driving distance. Also: I assume the earth is a sphere, so I don’t use geoid distances.
The gsl_siman_solve routine finds a route which is 3490.62
Kilometers long; this is confirmed by an exhaustive search of all
possible routes with the same initial city.
The full code can be found in siman/siman_tsp.c, but I include here some plots generated in the following way:
$ ./siman_tsp > tsp.output
$ grep -v "^#" tsp.output
| awk '{print $1, $NF}'
| graph -y 3300 6500 -W0 -X generation -Y distance
-L "TSP - 12 southwest cities"
| plot -Tps > 12-cities.eps
$ grep initial_city_coord tsp.output
| awk '{print $2, $3}'
| graph -X "longitude (- means west)" -Y "latitude"
-L "TSP - initial-order" -f 0.03 -S 1 0.1
| plot -Tps > initial-route.eps
$ grep final_city_coord tsp.output
| awk '{print $2, $3}'
| graph -X "longitude (- means west)" -Y "latitude"
-L "TSP - final-order" -f 0.03 -S 1 0.1
| plot -Tps > final-route.eps
This is the output showing the initial order of the cities; longitude is negative, since it is west and I want the plot to look like a map.
# initial coordinates of cities (longitude and latitude) ###initial_city_coord: -105.95 35.68 Santa Fe ###initial_city_coord: -112.07 33.54 Phoenix ###initial_city_coord: -106.62 35.12 Albuquerque ###initial_city_coord: -103.2 34.41 Clovis ###initial_city_coord: -107.87 37.29 Durango ###initial_city_coord: -96.77 32.79 Dallas ###initial_city_coord: -105.92 35.77 Tesuque ###initial_city_coord: -107.84 35.15 Grants ###initial_city_coord: -106.28 35.89 Los Alamos ###initial_city_coord: -106.76 32.34 Las Cruces ###initial_city_coord: -108.58 37.35 Cortez ###initial_city_coord: -108.74 35.52 Gallup ###initial_city_coord: -105.95 35.68 Santa Fe
The optimal route turns out to be:
# final coordinates of cities (longitude and latitude) ###final_city_coord: -105.95 35.68 Santa Fe ###final_city_coord: -103.2 34.41 Clovis ###final_city_coord: -96.77 32.79 Dallas ###final_city_coord: -106.76 32.34 Las Cruces ###final_city_coord: -112.07 33.54 Phoenix ###final_city_coord: -108.74 35.52 Gallup ###final_city_coord: -108.58 37.35 Cortez ###final_city_coord: -107.87 37.29 Durango ###final_city_coord: -107.84 35.15 Grants ###final_city_coord: -106.62 35.12 Albuquerque ###final_city_coord: -106.28 35.89 Los Alamos ###final_city_coord: -105.92 35.77 Tesuque ###final_city_coord: -105.95 35.68 Santa Fe
Here’s a plot of the cost function (energy) versus generation (point in the calculation at which a new temperature is set) for this problem:
Previous: Trivial example, Up: Examples with Simulated Annealing [Index]
Next: Root Finding Algorithms using Derivatives, Previous: Search Stopping Parameters, Up: One dimensional Root-Finding [Index]
The root bracketing algorithms described in this section require an initial interval which is guaranteed to contain a root—if a and b are the endpoints of the interval then f(a) must differ in sign from f(b). This ensures that the function crosses zero at least once in the interval. If a valid initial interval is used then these algorithm cannot fail, provided the function is well-behaved.
Note that a bracketing algorithm cannot find roots of even degree, since these do not cross the x-axis.
The bisection algorithm is the simplest method of bracketing the roots of a function. It is the slowest algorithm provided by the library, with linear convergence.
On each iteration, the interval is bisected and the value of the function at the midpoint is calculated. The sign of this value is used to determine which half of the interval does not contain a root. That half is discarded to give a new, smaller interval containing the root. This procedure can be continued indefinitely until the interval is sufficiently small.
At any time the current estimate of the root is taken as the midpoint of the interval.
The false position algorithm is a method of finding roots based on linear interpolation. Its convergence is linear, but it is usually faster than bisection.
On each iteration a line is drawn between the endpoints (a,f(a)) and (b,f(b)) and the point where this line crosses the x-axis taken as a “midpoint”. The value of the function at this point is calculated and its sign is used to determine which side of the interval does not contain a root. That side is discarded to give a new, smaller interval containing the root. This procedure can be continued indefinitely until the interval is sufficiently small.
The best estimate of the root is taken from the linear interpolation of the interval on the current iteration.
The Brent-Dekker method (referred to here as Brent’s method) combines an interpolation strategy with the bisection algorithm. This produces a fast algorithm which is still robust.
On each iteration Brent’s method approximates the function using an interpolating curve. On the first iteration this is a linear interpolation of the two endpoints. For subsequent iterations the algorithm uses an inverse quadratic fit to the last three points, for higher accuracy. The intercept of the interpolating curve with the x-axis is taken as a guess for the root. If it lies within the bounds of the current interval then the interpolating point is accepted, and used to generate a smaller interval. If the interpolating point is not accepted then the algorithm falls back to an ordinary bisection step.
The best estimate of the root is taken from the most recent interpolation or bisection.
Next: Root Finding Algorithms using Derivatives, Previous: Search Stopping Parameters, Up: One dimensional Root-Finding [Index]
Next: Error Codes, Up: Error Handling [Index]
The library follows the thread-safe error reporting conventions of the
POSIX Threads library. Functions return a non-zero error code to
indicate an error and 0 to indicate success.
int status = gsl_function (...)
if (status) { /* an error occurred */
.....
/* status value specifies the type of error */
}
The routines report an error whenever they cannot perform the task requested of them. For example, a root-finding function would return a non-zero error code if could not converge to the requested accuracy, or exceeded a limit on the number of iterations. Situations like this are a normal occurrence when using any mathematical library and you should check the return status of the functions that you call.
Whenever a routine reports an error the return value specifies the type
of error. The return value is analogous to the value of the variable
errno in the C library. The caller can examine the return code
and decide what action to take, including ignoring the error if it is
not considered serious.
In addition to reporting errors by return codes the library also has an
error handler function gsl_error. This function is called by
other library functions when they report an error, just before they
return to the caller. The default behavior of the error handler is to
print a message and abort the program,
gsl: file.c:67: ERROR: invalid argument supplied by user Default GSL error handler invoked. Aborted
The purpose of the gsl_error handler is to provide a function
where a breakpoint can be set that will catch library errors when
running under the debugger. It is not intended for use in production
programs, which should handle any errors using the return codes.
Next: Error Codes, Up: Error Handling [Index]
Next: Hurwitz Zeta Function, Previous: Riemann Zeta Function, Up: Zeta Functions [Index]
For large positive argument, the Riemann zeta function approaches one. In this region the fractional part is interesting, and therefore we need a function to evaluate it explicitly.
These routines compute \zeta(n) - 1 for integer n, n \ne 1.
These routines compute \zeta(s) - 1 for arbitrary s, s \ne 1.
Previous: Type Index, Up: Top [Index]
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Previous: Type Index, Up: Top [Index]
Next: 2D Interpolation Grids, Previous: 2D Introduction to Interpolation, Up: Interpolation [Index]
The interpolation function for a given dataset is stored in a
gsl_interp2d object. These are created by the following functions.
This function returns a pointer to a newly allocated interpolation object of type T for xsize grid points in the x direction and ysize grid points in the y direction.
This function initializes the interpolation object interp for the
data (xa,ya,za) where xa and ya are arrays of
the x and y grid points of size xsize and ysize
respectively, and za is an array of function values of size
xsize*ysize. The interpolation object (gsl_interp2d) does
not save the data arrays xa, ya, and za and only stores the
static state computed from the data. The xa and ya data arrays
are always assumed to be strictly ordered, with increasing x,y values;
the behavior for other arrangements is not defined.
This function frees the interpolation object interp.
Next: The gsl_sf_result struct, Up: Special Functions [Index]
The special functions are available in two calling conventions, a natural form which returns the numerical value of the function and an error-handling form which returns an error code. The two types of function provide alternative ways of accessing the same underlying code.
The natural form returns only the value of the function and can be used directly in mathematical expressions. For example, the following function call will compute the value of the Bessel function J_0(x),
double y = gsl_sf_bessel_J0 (x);
There is no way to access an error code or to estimate the error using this method. To allow access to this information the alternative error-handling form stores the value and error in a modifiable argument,
gsl_sf_result result; int status = gsl_sf_bessel_J0_e (x, &result);
The error-handling functions have the suffix _e. The returned
status value indicates error conditions such as overflow, underflow or
loss of precision. If there are no errors the error-handling functions
return GSL_SUCCESS.
Next: Stepping Functions, Up: Ordinary Differential Equations [Index]
The routines solve the general n-dimensional first-order system,
dy_i(t)/dt = f_i(t, y_1(t), ..., y_n(t))
for i = 1, \dots, n. The stepping functions rely on the vector
of derivatives f_i and the Jacobian matrix,
J_{ij} = df_i(t,y(t)) / dy_j.
A system of equations is defined using the gsl_odeiv2_system
datatype.
This data type defines a general ODE system with arbitrary parameters.
int (* function) (double t, const double y[], double dydt[], void * params)This function should store the vector elements f_i(t,y,params) in the array dydt, for arguments (t,y) and parameters params.
The function should return GSL_SUCCESS if the calculation was
completed successfully. Any other return value indicates an error. A
special return value GSL_EBADFUNC causes gsl_odeiv2
routines to immediately stop and return. If function
is modified (for example contents of params), the user must call an
appropriate reset function (gsl_odeiv2_driver_reset,
gsl_odeiv2_evolve_reset or gsl_odeiv2_step_reset)
before continuing. Use return values
distinct from standard GSL error codes to distinguish your function as
the source of the error.
int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void * params);This function should store the vector of derivative elements
in the array dfdt and the Jacobian matrix J_{ij} in the array dfdy, regarded as a row-ordered
matrix J(i,j) = dfdy[i * dimension + j] where dimension
is the dimension of the system.
Not all of the stepper algorithms of gsl_odeiv2 make use of the
Jacobian matrix, so it may not be necessary to provide this function
(the jacobian element of the struct can be replaced by a null
pointer for those algorithms).
The function should return GSL_SUCCESS if the calculation was
completed successfully. Any other return value indicates an error. A
special return value GSL_EBADFUNC causes gsl_odeiv2
routines to immediately stop and return. If jacobian
is modified (for example contents of params), the user must call an
appropriate reset function (gsl_odeiv2_driver_reset,
gsl_odeiv2_evolve_reset or gsl_odeiv2_step_reset)
before continuing. Use return values
distinct from standard GSL error codes to distinguish your function as
the source of the error.
size_t dimension;This is the dimension of the system of equations.
void * paramsThis is a pointer to the arbitrary parameters of the system.
Next: Stepping Functions, Up: Ordinary Differential Equations [Index]
Previous: Random Number Distribution Examples, Up: Random Number Distributions [Index]
For an encyclopaedic coverage of the subject readers are advised to consult the book Non-Uniform Random Variate Generation by Luc Devroye. It covers every imaginable distribution and provides hundreds of algorithms.
The subject of random variate generation is also reviewed by Knuth, who describes algorithms for all the major distributions.
The Particle Data Group provides a short review of techniques for generating distributions of random numbers in the “Monte Carlo” section of its Annual Review of Particle Physics.
The Review of Particle Physics is available online in postscript and pdf format.
An overview of methods used to compute cumulative distribution functions can be found in Statistical Computing by W.J. Kennedy and J.E. Gentle. Another general reference is Elements of Statistical Computing by R.A. Thisted.
The cumulative distribution functions for the Gaussian distribution are based on the following papers,
Previous: Random Number Distribution Examples, Up: Random Number Distributions [Index]
The first example, in one dimensional Cartesian space, sets up an energy function which is a damped sine wave; this has many local minima, but only one global minimum, somewhere between 1.0 and 1.5. The initial guess given is 15.5, which is several local minima away from the global minimum.
#include <math.h>
#include <stdlib.h>
#include <string.h>
#include <gsl/gsl_siman.h>
/* set up parameters for this simulated annealing run */
/* how many points do we try before stepping */
#define N_TRIES 200
/* how many iterations for each T? */
#define ITERS_FIXED_T 1000
/* max step size in random walk */
#define STEP_SIZE 1.0
/* Boltzmann constant */
#define K 1.0
/* initial temperature */
#define T_INITIAL 0.008
/* damping factor for temperature */
#define MU_T 1.003
#define T_MIN 2.0e-6
gsl_siman_params_t params
= {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
K, T_INITIAL, MU_T, T_MIN};
/* now some functions to test in one dimension */
double E1(void *xp)
{
double x = * ((double *) xp);
return exp(-pow((x-1.0),2.0))*sin(8*x);
}
double M1(void *xp, void *yp)
{
double x = *((double *) xp);
double y = *((double *) yp);
return fabs(x - y);
}
void S1(const gsl_rng * r, void *xp, double step_size)
{
double old_x = *((double *) xp);
double new_x;
double u = gsl_rng_uniform(r);
new_x = u * 2 * step_size - step_size + old_x;
memcpy(xp, &new_x, sizeof(new_x));
}
void P1(void *xp)
{
printf ("%12g", *((double *) xp));
}
int
main(void)
{
const gsl_rng_type * T;
gsl_rng * r;
double x_initial = 15.5;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc(T);
gsl_siman_solve(r, &x_initial, E1, S1, M1, P1,
NULL, NULL, NULL,
sizeof(double), params);
gsl_rng_free (r);
return 0;
}
Here are a couple of plots that are generated by running
siman_test in the following way:
$ ./siman_test | awk '!/^#/ {print $1, $4}'
| graph -y 1.34 1.4 -W0 -X generation -Y position
| plot -Tps > siman-test.eps
$ ./siman_test | awk '!/^#/ {print $1, $5}'
| graph -y -0.88 -0.83 -W0 -X generation -Y energy
| plot -Tps > siman-energy.eps
Next: Beta Functions, Previous: Pochhammer Symbol, Up: Gamma and Beta Functions [Index]
These functions compute the unnormalized incomplete Gamma Function \Gamma(a,x) = \int_x^\infty dt t^{a-1} \exp(-t) for a real and x >= 0.
These routines compute the normalized incomplete Gamma Function Q(a,x) = 1/\Gamma(a) \int_x^\infty dt t^{a-1} \exp(-t) for a > 0, x >= 0.
These routines compute the complementary normalized incomplete Gamma Function P(a,x) = 1 - Q(a,x) = 1/\Gamma(a) \int_0^x dt t^{a-1} \exp(-t) for a > 0, x >= 0.
Note that Abramowitz & Stegun call P(a,x) the incomplete gamma function (section 6.5).
Next: Series Acceleration, Previous: Numerical Differentiation, Up: Top [Index]
This chapter describes routines for computing Chebyshev approximations to univariate functions. A Chebyshev approximation is a truncation of the series f(x) = \sum c_n T_n(x), where the Chebyshev polynomials T_n(x) = \cos(n \arccos x) provide an orthogonal basis of polynomials on the interval [-1,1] with the weight function 1 / \sqrt{1-x^2}. The first few Chebyshev polynomials are, T_0(x) = 1, T_1(x) = x, T_2(x) = 2 x^2 - 1. For further information see Abramowitz & Stegun, Chapter 22.
The functions described in this chapter are declared in the header file gsl_chebyshev.h.
The following functions compute the full Levin u-transform of a series with its error estimate. The error estimate is computed by propagating rounding errors from each term through to the final extrapolation.
These functions are intended for summing analytic series where each term
is known to high accuracy, and the rounding errors are assumed to
originate from finite precision. They are taken to be relative errors of
order GSL_DBL_EPSILON for each term.
The calculation of the error in the extrapolated value is an O(N^2) process, which is expensive in time and memory. A faster but less reliable method which estimates the error from the convergence of the extrapolated value is described in the next section. For the method described here a full table of intermediate values and derivatives through to O(N) must be computed and stored, but this does give a reliable error estimate.
This function allocates a workspace for a Levin u-transform of n terms. The size of the workspace is O(2n^2 + 3n).
This function frees the memory associated with the workspace w.
This function takes the terms of a series in array of size
array_size and computes the extrapolated limit of the series using
a Levin u-transform. Additional working space must be provided in
w. The extrapolated sum is stored in sum_accel, with an
estimate of the absolute error stored in abserr. The actual
term-by-term sum is returned in w->sum_plain. The algorithm
calculates the truncation error (the difference between two successive
extrapolations) and round-off error (propagated from the individual
terms) to choose an optimal number of terms for the extrapolation.
All the terms of the series passed in through array should be non-zero.
Next: Fitting References and Further Reading, Previous: Troubleshooting, Up: Least-Squares Fitting [Index]
The example programs in this section demonstrate the various linear regression methods.
Next: Tridiagonal Decomposition of Hermitian Matrices, Previous: Modified Cholesky Decomposition, Up: Linear Algebra [Index]
A symmetric matrix A can be factorized by similarity transformations into the form,
A = Q T Q^T
where Q is an orthogonal matrix and T is a symmetric tridiagonal matrix.
This function factorizes the symmetric square matrix A into the symmetric tridiagonal decomposition Q T Q^T. On output the diagonal and subdiagonal part of the input matrix A contain the tridiagonal matrix T. The remaining lower triangular part of the input matrix contains the Householder vectors which, together with the Householder coefficients tau, encode the orthogonal matrix Q. This storage scheme is the same as used by LAPACK. The upper triangular part of A is not referenced.
This function unpacks the encoded symmetric tridiagonal decomposition
(A, tau) obtained from gsl_linalg_symmtd_decomp into
the orthogonal matrix Q, the vector of diagonal elements diag
and the vector of subdiagonal elements subdiag.
This function unpacks the diagonal and subdiagonal of the encoded
symmetric tridiagonal decomposition (A, tau) obtained from
gsl_linalg_symmtd_decomp into the vectors diag and subdiag.
Next: The Logarithmic Distribution, Previous: The Geometric Distribution, Up: Random Number Distributions [Index]
This function returns a random integer from the hypergeometric distribution. The probability distribution for hypergeometric random variates is,
p(k) = C(n_1, k) C(n_2, t - k) / C(n_1 + n_2, t)
where C(a,b) = a!/(b!(a-b)!) and t <= n_1 + n_2. The domain of k is max(0,t-n_2), ..., min(t,n_1).
If a population contains n_1 elements of “type 1” and n_2 elements of “type 2” then the hypergeometric distribution gives the probability of obtaining k elements of “type 1” in t samples from the population without replacement.
This function computes the probability p(k) of obtaining k from a hypergeometric distribution with parameters n1, n2, t, using the formula given above.
These functions compute the cumulative distribution functions P(k), Q(k) for the hypergeometric distribution with parameters n1, n2 and t.
Next: 2D Interpolation Example programs, Previous: 2D Evaluation of Interpolating Functions, Up: Interpolation [Index]
The functions described in the previous sections required the user to
supply pointers to the x, y, and z arrays on each call.
The following functions are equivalent to the corresponding
gsl_interp2d functions but maintain a copy of this data in the
gsl_spline2d object. This removes the need to pass xa,
ya, and za as arguments on each evaluation. These functions are
defined in the header file gsl_spline2d.h.
This function returns the value z_{ij} for grid point (i,j) stored in the array za.
Next: 2D Interpolation Example programs, Previous: 2D Evaluation of Interpolating Functions, Up: Interpolation [Index]
Next: Transport Functions, Previous: Psi (Digamma) Function, Up: Special Functions [Index]
The functions described in this section are declared in the header file gsl_sf_synchrotron.h.
These routines compute the first synchrotron function x \int_x^\infty dt K_{5/3}(t) for x >= 0.
These routines compute the second synchrotron function x K_{2/3}(x) for x >= 0.
Next: Multimin Algorithms without Derivatives, Previous: Multimin Stopping Criteria, Up: Multidimensional Minimization [Index]
There are several minimization methods available. The best choice of algorithm depends on the problem. The algorithms described in this section use the value of the function and its gradient at each evaluation point.
This is the Fletcher-Reeves conjugate gradient algorithm. The conjugate gradient algorithm proceeds as a succession of line minimizations. The sequence of search directions is used to build up an approximation to the curvature of the function in the neighborhood of the minimum.
An initial search direction p is chosen using the gradient, and line minimization is carried out in that direction. The accuracy of the line minimization is specified by the parameter tol. The minimum along this line occurs when the function gradient g and the search direction p are orthogonal. The line minimization terminates when dot(p,g) < tol |p| |g|. The search direction is updated using the Fletcher-Reeves formula p' = g' - \beta g where \beta=-|g'|^2/|g|^2, and the line minimization is then repeated for the new search direction.
This is the Polak-Ribiere conjugate gradient algorithm. It is similar to the Fletcher-Reeves method, differing only in the choice of the coefficient \beta. Both methods work well when the evaluation point is close enough to the minimum of the objective function that it is well approximated by a quadratic hypersurface.
These methods use the vector Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. This is a quasi-Newton method which builds up an approximation to the second derivatives of the function f using the difference between successive gradient vectors. By combining the first and second derivatives the algorithm is able to take Newton-type steps towards the function minimum, assuming quadratic behavior in that region.
The bfgs2 version of this minimizer is the most efficient
version available, and is a faithful implementation of the line
minimization scheme described in Fletcher’s Practical Methods of
Optimization, Algorithms 2.6.2 and 2.6.4. It supersedes the original
bfgs routine and requires substantially fewer function and
gradient evaluations. The user-supplied tolerance tol
corresponds to the parameter \sigma used by Fletcher. A value
of 0.1 is recommended for typical use (larger values correspond to
less accurate line searches).
The steepest descent algorithm follows the downhill gradient of the function at each step. When a downhill step is successful the step-size is increased by a factor of two. If the downhill step leads to a higher function value then the algorithm backtracks and the step size is decreased using the parameter tol. A suitable value of tol for most applications is 0.1. The steepest descent method is inefficient and is included only for demonstration purposes.
Next: Multimin Algorithms without Derivatives, Previous: Multimin Stopping Criteria, Up: Multidimensional Minimization [Index]
Previous: Quasi-random number generator examples, Up: Quasi-Random Sequences [Index]
The implementations of the quasi-random sequence routines are based on the algorithms described in the following paper,
Next: The Multivariate Gaussian Distribution, Previous: The Gaussian Tail Distribution, Up: Random Number Distributions [Index]
This function generates a pair of correlated Gaussian variates, with mean zero, correlation coefficient rho and standard deviations sigma_x and sigma_y in the x and y directions. The probability distribution for bivariate Gaussian random variates is,
p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy
for x,y in the range -\infty to +\infty. The correlation coefficient rho should lie between 1 and -1.
This function computes the probability density p(x,y) at (x,y) for a bivariate Gaussian distribution with standard deviations sigma_x, sigma_y and correlation coefficient rho, using the formula given above.
Next: The Multinomial Distribution, Previous: The Bernoulli Distribution, Up: Random Number Distributions [Index]
This function returns a random integer from the binomial distribution, the number of successes in n independent trials with probability p. The probability distribution for binomial variates is,
p(k) = {n! \over k! (n-k)! } p^k (1-p)^{n-k}
for 0 <= k <= n.
This function computes the probability p(k) of obtaining k from a binomial distribution with parameters p and n, using the formula given above.
These functions compute the cumulative distribution functions P(k), Q(k) for the binomial distribution with parameters p and n.
Next: Multidimensional Minimization, Previous: One dimensional Minimization, Up: Top [Index]
This chapter describes functions for multidimensional root-finding (solving nonlinear systems with n equations in n unknowns). The library provides low level components for a variety of iterative solvers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the iteration. Each class of methods uses the same framework, so that you can switch between solvers at runtime without needing to recompile your program. Each instance of a solver keeps track of its own state, allowing the solvers to be used in multi-threaded programs. The solvers are based on the original Fortran library MINPACK.
The header file gsl_multiroots.h contains prototypes for the multidimensional root finding functions and related declarations.
Next: Reading and writing combinations, Previous: Combination properties, Up: Combinations [Index]
This function advances the combination c to the next combination
in lexicographic order and returns GSL_SUCCESS. If no further
combinations are available it returns GSL_FAILURE and leaves
c unmodified. Starting with the first combination and
repeatedly applying this function will iterate through all possible
combinations of a given order.
This function steps backwards from the combination c to the
previous combination in lexicographic order, returning
GSL_SUCCESS. If no previous combination is available it returns
GSL_FAILURE and leaves c unmodified.
Next: Givens Rotations, Previous: Hessenberg-Triangular Decomposition of Real Matrices, Up: Linear Algebra [Index]
A general matrix A can be factorized by similarity transformations into the form,
A = U B V^T
where U and V are orthogonal matrices and B is a N-by-N bidiagonal matrix with non-zero entries only on the diagonal and superdiagonal. The size of U is M-by-N and the size of V is N-by-N.
This function factorizes the M-by-N matrix A into bidiagonal form U B V^T. The diagonal and superdiagonal of the matrix B are stored in the diagonal and superdiagonal of A. The orthogonal matrices U and V are stored as compressed Householder vectors in the remaining elements of A. The Householder coefficients are stored in the vectors tau_U and tau_V. The length of tau_U must equal the number of elements in the diagonal of A and the length of tau_V should be one element shorter.
This function unpacks the bidiagonal decomposition of A produced by
gsl_linalg_bidiag_decomp, (A, tau_U, tau_V)
into the separate orthogonal matrices U, V and the diagonal
vector diag and superdiagonal superdiag. Note that U
is stored as a compact M-by-N orthogonal matrix satisfying
U^T U = I for efficiency.
This function unpacks the bidiagonal decomposition of A produced by
gsl_linalg_bidiag_decomp, (A, tau_U, tau_V)
into the separate orthogonal matrices U, V and the diagonal
vector diag and superdiagonal superdiag. The matrix U
is stored in-place in A.
This function unpacks the diagonal and superdiagonal of the bidiagonal
decomposition of A from gsl_linalg_bidiag_decomp, into
the diagonal vector diag and superdiagonal vector superdiag.
Next: Givens Rotations, Previous: Hessenberg-Triangular Decomposition of Real Matrices, Up: Linear Algebra [Index]
This function returns a pointer to a newly-created instance of a
quasi-random sequence generator of type T and dimension d.
If there is insufficient memory to create the generator then the
function returns a null pointer and the error handler is invoked with an
error code of GSL_ENOMEM.
This function frees all the memory associated with the generator q.
This function reinitializes the generator q to its starting point. Note that quasi-random sequences do not use a seed and always produce the same set of values.
Next: Providing a function to minimize, Previous: Multimin Caveats, Up: Multidimensional Minimization [Index]
The following function initializes a multidimensional minimizer. The minimizer itself depends only on the dimension of the problem and the algorithm and can be reused for different problems.
This function returns a pointer to a newly allocated instance of a
minimizer of type T for an n-dimension function. If there
is insufficient memory to create the minimizer then the function returns
a null pointer and the error handler is invoked with an error code of
GSL_ENOMEM.
The function gsl_multimin_fdfminimizer_set initializes the minimizer s to minimize the function
fdf starting from the initial point x. The size of the
first trial step is given by step_size. The accuracy of the line
minimization is specified by tol. The precise meaning of this
parameter depends on the method used. Typically the line minimization
is considered successful if the gradient of the function g is
orthogonal to the current search direction p to a relative
accuracy of tol, where dot(p,g) < tol |p| |g|. A tol value of 0.1 is
suitable for most purposes, since line minimization only needs to
be carried out approximately. Note that setting tol to zero will
force the use of “exact” line-searches, which are extremely expensive.
The function gsl_multimin_fminimizer_set initializes the minimizer s to minimize the function
f, starting from the initial point
x. The size of the initial trial steps is given in vector
step_size. The precise meaning of this parameter depends on the
method used.
This function frees all the memory associated with the minimizer s.
This function returns a pointer to the name of the minimizer. For example,
printf ("s is a '%s' minimizer\n",
gsl_multimin_fdfminimizer_name (s));
would print something like s is a 'conjugate_pr' minimizer.
Next: Providing a function to minimize, Previous: Multimin Caveats, Up: Multidimensional Minimization [Index]
The discrete Hankel transform acts on a vector of sampled data, where the samples are assumed to have been taken at points related to the zeros of a Bessel function of fixed order; compare this to the case of the discrete Fourier transform, where samples are taken at points related to the zeroes of the sine or cosine function.
Starting with its definition, the Hankel transform (or Bessel transform) of order \nu of a function f with \nu > -1/2 is defined as (see Johnson, 1987 and Lemoine, 1994)
F_\nu(u) = \int_0^\infty f(t) J_\nu(u t) t dt
If the integral exists, F_\nu is called the Hankel transformation of f. The reverse transform is given by
f(t) = \int_0^\infty F_\nu(u) J_\nu(u t) u du ,
where \int_0^\infty f(t) t^{1/2} dt must exist and be absolutely convergent, and where f(t) satisfies Dirichlet’s conditions (of limited total fluctuations) in the interval [0,\infty].
Now the discrete Hankel transform works on a discrete function f, which is sampled on points n=1...M located at positions t_n=(j_{\nu,n}/j_{\nu,M}) X in real space and at u_n=j_{\nu,n}/X in reciprocal space. Here, j_{\nu,m} are the m-th zeros of the Bessel function J_\nu(x) arranged in ascending order. Moreover, the discrete functions are assumed to be band limited, so f(t_n)=0 and F(u_n)=0 for n>M. Accordingly, the function f is defined on the interval [0,X].
Following the work of Johnson, 1987 and Lemoine, 1994, the discrete Hankel transform is given by
F_\nu(u_m) = (2 X^2 / j_(\nu,M)^2)
\sum_{k=1}^{M-1} f(j_(\nu,k) X/j_(\nu,M))
(J_\nu(j_(\nu,m) j_(\nu,k) / j_(\nu,M)) / J_(\nu+1)(j_(\nu,k))^2).
It is this discrete expression which defines the discrete Hankel transform calculated by GSL. In GSL, forward and backward transforms are defined equally and calculate F_\nu(u_m). Following Johnson, the backward transform reads
f(t_k) = (2 / X^2)
\sum_{m=1}^{M-1} F(j_(\nu,m)/X)
(J_\nu(j_(\nu,m) j_(\nu,k) / j_(\nu,M)) / J_(\nu+1)(j_(\nu,m))^2).
Obviously, using the forward transform instead of the backward transform gives an additional factor X^4/j_{\nu,M}^2=t_m^2/u_m^2.
The kernel in the summation above defines the matrix of the
\nu-Hankel transform of size M-1. The coefficients of
this matrix, being dependent on \nu and M, must be
precomputed and stored; the gsl_dht object encapsulates this
data. The allocation function gsl_dht_alloc returns a
gsl_dht object which must be properly initialized with
gsl_dht_init before it can be used to perform transforms on data
sample vectors, for fixed \nu and M, using the
gsl_dht_apply function. The implementation allows to define the
length X of the fundamental interval, for convenience, while
discrete Hankel transforms are often defined on the unit interval
instead of [0,X].
Notice that by assumption f(t) vanishes at the endpoints of the interval, consistent with the inversion formula and the sampling formula given above. Therefore, this transform corresponds to an orthogonal expansion in eigenfunctions of the Dirichlet problem for the Bessel differential equation.
Previous: Vector properties, Up: Vectors [Index]
This program shows how to allocate, initialize and read from a vector
using the functions gsl_vector_alloc, gsl_vector_set and
gsl_vector_get.
#include <stdio.h>
#include <gsl/gsl_vector.h>
int
main (void)
{
int i;
gsl_vector * v = gsl_vector_alloc (3);
for (i = 0; i < 3; i++)
{
gsl_vector_set (v, i, 1.23 + i);
}
for (i = 0; i < 100; i++) /* OUT OF RANGE ERROR */
{
printf ("v_%d = %g\n", i, gsl_vector_get (v, i));
}
gsl_vector_free (v);
return 0;
}
Here is the output from the program. The final loop attempts to read
outside the range of the vector v, and the error is trapped by
the range-checking code in gsl_vector_get.
$ ./a.out v_0 = 1.23 v_1 = 2.23 v_2 = 3.23 gsl: vector_source.c:12: ERROR: index out of range Default GSL error handler invoked. Aborted (core dumped)
The next program shows how to write a vector to a file.
#include <stdio.h>
#include <gsl/gsl_vector.h>
int
main (void)
{
int i;
gsl_vector * v = gsl_vector_alloc (100);
for (i = 0; i < 100; i++)
{
gsl_vector_set (v, i, 1.23 + i);
}
{
FILE * f = fopen ("test.dat", "w");
gsl_vector_fprintf (f, v, "%.5g");
fclose (f);
}
gsl_vector_free (v);
return 0;
}
After running this program the file test.dat should contain the
elements of v, written using the format specifier
%.5g. The vector could then be read back in using the function
gsl_vector_fscanf (f, v) as follows:
#include <stdio.h>
#include <gsl/gsl_vector.h>
int
main (void)
{
int i;
gsl_vector * v = gsl_vector_alloc (10);
{
FILE * f = fopen ("test.dat", "r");
gsl_vector_fscanf (f, v);
fclose (f);
}
for (i = 0; i < 10; i++)
{
printf ("%g\n", gsl_vector_get(v, i));
}
gsl_vector_free (v);
return 0;
}
Next: VEGAS, Previous: PLAIN Monte Carlo, Up: Monte Carlo Integration [Index]
The MISER algorithm of Press and Farrar is based on recursive stratified sampling. This technique aims to reduce the overall integration error by concentrating integration points in the regions of highest variance.
The idea of stratified sampling begins with the observation that for two disjoint regions a and b with Monte Carlo estimates of the integral E_a(f) and E_b(f) and variances \sigma_a^2(f) and \sigma_b^2(f), the variance \Var(f) of the combined estimate E(f) = (1/2) (E_a(f) + E_b(f)) is given by,
\Var(f) = (\sigma_a^2(f) / 4 N_a) + (\sigma_b^2(f) / 4 N_b).
It can be shown that this variance is minimized by distributing the points such that,
N_a / (N_a + N_b) = \sigma_a / (\sigma_a + \sigma_b).
Hence the smallest error estimate is obtained by allocating sample points in proportion to the standard deviation of the function in each sub-region.
The MISER algorithm proceeds by bisecting the integration region along one coordinate axis to give two sub-regions at each step. The direction is chosen by examining all d possible bisections and selecting the one which will minimize the combined variance of the two sub-regions. The variance in the sub-regions is estimated by sampling with a fraction of the total number of points available to the current step. The same procedure is then repeated recursively for each of the two half-spaces from the best bisection. The remaining sample points are allocated to the sub-regions using the formula for N_a and N_b. This recursive allocation of integration points continues down to a user-specified depth where each sub-region is integrated using a plain Monte Carlo estimate. These individual values and their error estimates are then combined upwards to give an overall result and an estimate of its error.
The functions described in this section are declared in the header file gsl_monte_miser.h.
This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions. The workspace is used to maintain the state of the integration.
This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.
This routines uses the MISER Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses a fixed number of function calls calls, and obtains random sampling points using the random number generator r. A previously allocated workspace s must be supplied. The result of the integration is returned in result, with an estimated absolute error abserr.
This function frees the memory associated with the integrator state s.
The MISER algorithm has several configurable parameters which can be changed using the following two functions.13
This function copies the parameters of the integrator state into the user-supplied params structure.
This function sets the integrator parameters based on values provided in the params structure.
Typically the values of the parameters are first read using
gsl_monte_miser_params_get, the necessary changes are made to
the fields of the params structure, and the values are copied
back into the integrator state using
gsl_monte_miser_params_set. The functions use the
gsl_monte_miser_params structure which contains the following
fields:
This parameter specifies the fraction of the currently available number of function calls which are allocated to estimating the variance at each recursive step. The default value is 0.1.
This parameter specifies the minimum number of function calls required
for each estimate of the variance. If the number of function calls
allocated to the estimate using estimate_frac falls below
min_calls then min_calls are used instead. This ensures
that each estimate maintains a reasonable level of accuracy. The
default value of min_calls is 16 * dim.
This parameter specifies the minimum number of function calls required
to proceed with a bisection step. When a recursive step has fewer calls
available than min_calls_per_bisection it performs a plain Monte
Carlo estimate of the current sub-region and terminates its branch of
the recursion. The default value of this parameter is 32 *
min_calls.
This parameter controls how the estimated variances for the two sub-regions of a bisection are combined when allocating points. With recursive sampling the overall variance should scale better than 1/N, since the values from the sub-regions will be obtained using a procedure which explicitly minimizes their variance. To accommodate this behavior the MISER algorithm allows the total variance to depend on a scaling parameter \alpha,
\Var(f) = {\sigma_a \over N_a^\alpha} + {\sigma_b \over N_b^\alpha}.
The authors of the original paper describing MISER recommend the value \alpha = 2 as a good choice, obtained from numerical experiments, and this is used as the default value in this implementation.
This parameter introduces a random fractional variation of size dither into each bisection, which can be used to break the symmetry of integrands which are concentrated near the exact center of the hypercubic integration region. The default value of dither is zero, so no variation is introduced. If needed, a typical value of dither is 0.1.
The previous
method of accessing these fields directly through the
gsl_monte_miser_state struct is now deprecated.
Next: VEGAS, Previous: PLAIN Monte Carlo, Up: Monte Carlo Integration [Index]
Next: Radioactivity, Previous: Viscosity, Up: Physical Constants [Index]
GSL_CONST_MKSA_STILBThe luminance of 1 stilb.
GSL_CONST_MKSA_LUMENThe luminous flux of 1 lumen.
GSL_CONST_MKSA_LUXThe illuminance of 1 lux.
GSL_CONST_MKSA_PHOTThe illuminance of 1 phot.
GSL_CONST_MKSA_FOOTCANDLEThe illuminance of 1 footcandle.
GSL_CONST_MKSA_LAMBERTThe luminance of 1 lambert.
GSL_CONST_MKSA_FOOTLAMBERTThe luminance of 1 footlambert.
Next: Algorithms using Derivatives, Previous: Iteration of the multidimensional solver, Up: Multidimensional Root-Finding [Index]
A root finding procedure should stop when one of the following conditions is true:
The handling of these conditions is under user control. The functions below allow the user to test the precision of the current result in several standard ways.
This function tests for the convergence of the sequence by comparing the
last step dx with the absolute error epsabs and relative
error epsrel to the current position x. The test returns
GSL_SUCCESS if the following condition is achieved,
|dx_i| < epsabs + epsrel |x_i|
for each component of x and returns GSL_CONTINUE otherwise.
This function tests the residual value f against the absolute
error bound epsabs. The test returns GSL_SUCCESS if the
following condition is achieved,
\sum_i |f_i| < epsabs
and returns GSL_CONTINUE otherwise. This criterion is suitable
for situations where the precise location of the root, x, is
unimportant provided a value can be found where the residual is small
enough.
Next: Numerical Differentiation References, Previous: Numerical Differentiation functions, Up: Numerical Differentiation [Index]
The following code estimates the derivative of the function
f(x) = x^{3/2}
at x=2 and at x=0. The function f(x) is
undefined for x<0 so the derivative at x=0 is computed
using gsl_deriv_forward.
#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_deriv.h>
double f (double x, void * params)
{
(void)(params); /* avoid unused parameter warning */
return pow (x, 1.5);
}
int
main (void)
{
gsl_function F;
double result, abserr;
F.function = &f;
F.params = 0;
printf ("f(x) = x^(3/2)\n");
gsl_deriv_central (&F, 2.0, 1e-8, &result, &abserr);
printf ("x = 2.0\n");
printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
printf ("exact = %.10f\n\n", 1.5 * sqrt(2.0));
gsl_deriv_forward (&F, 0.0, 1e-8, &result, &abserr);
printf ("x = 0.0\n");
printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
printf ("exact = %.10f\n", 0.0);
return 0;
}
Here is the output of the program,
$ ./a.out
f(x) = x^(3/2) x = 2.0 f'(x) = 2.1213203120 +/- 0.0000005006 exact = 2.1213203436 x = 0.0 f'(x) = 0.0000000160 +/- 0.0000000339 exact = 0.0000000000
Next: The Type-1 Gumbel Distribution, Previous: Spherical Vector Distributions, Up: Random Number Distributions [Index]
This function returns a random variate from the Weibull distribution. The distribution function is,
p(x) dx = {b \over a^b} x^{b-1} \exp(-(x/a)^b) dx
for x >= 0.
This function computes the probability density p(x) at x for a Weibull distribution with scale a and exponent b, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Weibull distribution with scale a and exponent b.
Next: Working with the Greville abscissae, Previous: Evaluation of B-spline basis functions, Up: Basis Splines [Index]
This function evaluates all B-spline basis function derivatives of orders
0 through nderiv (inclusive) at the position x
and stores them in the matrix dB. The (i,j)-th element of dB
is d^jB_i(x)/dx^j. The matrix dB must be
of size n = nbreak + k - 2 by nderiv + 1.
The value n may also be obtained
by calling gsl_bspline_ncoeffs. Note that function evaluations
are included as the zeroth order derivatives in dB.
Computing all the basis function derivatives at once is more efficient
than computing them individually, due to the nature of the defining
recurrence relation.
This function evaluates all potentially nonzero B-spline basis function derivatives of orders 0 through nderiv (inclusive) at the position x and stores them in the matrix dB. The (i,j)-th element of dB is d^j/dx^j B_(istart+i)(x). The last row of dB contains d^j/dx^j B_(iend)(x). The matrix dB must be of size k by at least nderiv + 1. Note that function evaluations are included as the zeroth order derivatives in dB. By returning only the nonzero basis functions, this function allows quantities involving linear combinations of the B_i(x) and their derivatives to be computed without unnecessary terms.
Next: 1D Interpolation References and Further Reading, Previous: 1D Higher-level Interface, Up: Interpolation [Index]
The following program demonstrates the use of the interpolation and spline functions. It computes a cubic spline interpolation of the 10-point dataset (x_i, y_i) where x_i = i + \sin(i)/2 and y_i = i + \cos(i^2) for i = 0 \dots 9.
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_spline.h>
int
main (void)
{
int i;
double xi, yi, x[10], y[10];
printf ("#m=0,S=2\n");
for (i = 0; i < 10; i++)
{
x[i] = i + 0.5 * sin (i);
y[i] = i + cos (i * i);
printf ("%g %g\n", x[i], y[i]);
}
printf ("#m=1,S=0\n");
{
gsl_interp_accel *acc
= gsl_interp_accel_alloc ();
gsl_spline *spline
= gsl_spline_alloc (gsl_interp_cspline, 10);
gsl_spline_init (spline, x, y, 10);
for (xi = x[0]; xi < x[9]; xi += 0.01)
{
yi = gsl_spline_eval (spline, xi, acc);
printf ("%g %g\n", xi, yi);
}
gsl_spline_free (spline);
gsl_interp_accel_free (acc);
}
return 0;
}
The output is designed to be used with the GNU plotutils
graph program,
$ ./a.out > interp.dat $ graph -T ps < interp.dat > interp.ps
The result shows a smooth interpolation of the original points. The
interpolation method can be changed simply by varying the first argument of
gsl_spline_alloc.
The next program demonstrates a periodic cubic spline with 4 data points. Note that the first and last points must be supplied with the same y-value for a periodic spline.
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_spline.h>
int
main (void)
{
int N = 4;
double x[4] = {0.00, 0.10, 0.27, 0.30};
double y[4] = {0.15, 0.70, -0.10, 0.15};
/* Note: y[0] == y[3] for periodic data */
gsl_interp_accel *acc = gsl_interp_accel_alloc ();
const gsl_interp_type *t = gsl_interp_cspline_periodic;
gsl_spline *spline = gsl_spline_alloc (t, N);
int i; double xi, yi;
printf ("#m=0,S=5\n");
for (i = 0; i < N; i++)
{
printf ("%g %g\n", x[i], y[i]);
}
printf ("#m=1,S=0\n");
gsl_spline_init (spline, x, y, N);
for (i = 0; i <= 100; i++)
{
xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
yi = gsl_spline_eval (spline, xi, acc);
printf ("%g %g\n", xi, yi);
}
gsl_spline_free (spline);
gsl_interp_accel_free (acc);
return 0;
}
The output can be plotted with GNU graph.
$ ./a.out > interp.dat $ graph -T ps < interp.dat > interp.ps
The result shows a periodic interpolation of the original points. The slope of the fitted curve is the same at the beginning and end of the data, and the second derivative is also.
The next program illustrates the difference between the cubic spline, Akima, and Steffen interpolation types on a difficult dataset.
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_spline.h>
int
main(void)
{
size_t i;
const size_t N = 9;
/* this dataset is taken from
* J. M. Hyman, Accurate Monotonicity preserving cubic interpolation,
* SIAM J. Sci. Stat. Comput. 4, 4, 1983. */
const double x[] = { 7.99, 8.09, 8.19, 8.7, 9.2,
10.0, 12.0, 15.0, 20.0 };
const double y[] = { 0.0, 2.76429e-5, 4.37498e-2,
0.169183, 0.469428, 0.943740,
0.998636, 0.999919, 0.999994 };
gsl_interp_accel *acc = gsl_interp_accel_alloc();
gsl_spline *spline_cubic = gsl_spline_alloc(gsl_interp_cspline, N);
gsl_spline *spline_akima = gsl_spline_alloc(gsl_interp_akima, N);
gsl_spline *spline_steffen = gsl_spline_alloc(gsl_interp_steffen, N);
gsl_spline_init(spline_cubic, x, y, N);
gsl_spline_init(spline_akima, x, y, N);
gsl_spline_init(spline_steffen, x, y, N);
for (i = 0; i < N; ++i)
printf("%g %g\n", x[i], y[i]);
printf("\n\n");
for (i = 0; i <= 100; ++i)
{
double xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
double yi_cubic = gsl_spline_eval(spline_cubic, xi, acc);
double yi_akima = gsl_spline_eval(spline_akima, xi, acc);
double yi_steffen = gsl_spline_eval(spline_steffen, xi, acc);
printf("%g %g %g %g\n", xi, yi_cubic, yi_akima, yi_steffen);
}
gsl_spline_free(spline_cubic);
gsl_spline_free(spline_akima);
gsl_spline_free(spline_steffen);
gsl_interp_accel_free(acc);
return 0;
}
The cubic method exhibits a local maxima between the 6th and 7th data points and continues oscillating for the rest of the data. Akima also shows a local maxima but recovers and follows the data well after the 7th grid point. Steffen preserves monotonicity in all intervals and does not exhibit oscillations, at the expense of having a discontinuous second derivative.
Next: 1D Interpolation References and Further Reading, Previous: 1D Higher-level Interface, Up: Interpolation [Index]
Next: Vector operations, Previous: Copying vectors, Up: Vectors [Index]
The following function can be used to exchange, or permute, the elements of a vector.
This function exchanges the i-th and j-th elements of the vector v in-place.
This function reverses the order of the elements of the vector v.
Next: Special Functions, Previous: Complex Numbers, Up: Top [Index]
This chapter describes functions for evaluating and solving polynomials. There are routines for finding real and complex roots of quadratic and cubic equations using analytic methods. An iterative polynomial solver is also available for finding the roots of general polynomials with real coefficients (of any order). The functions are declared in the header file gsl_poly.h.
Previous: ODE Example programs, Up: Ordinary Differential Equations [Index]
Many of the basic Runge-Kutta formulas can be found in the Handbook of Mathematical Functions,
The implicit Bulirsch-Stoer algorithm bsimp is described in the
following paper,
The Adams and BDF multistep methods msadams and msbdf
are based on the following articles,
Next: 2D Interpolation Types, Previous: 2D Interpolation Functions, Up: Interpolation [Index]
The 2D interpolation routines access the function values z_{ij} with the following ordering:
z_ij = za[j*xsize + i]
with i = 0,...,xsize-1 and j = 0,...,ysize-1. However, for ease of use, the following functions are provided to add and retrieve elements from the function grid without requiring knowledge of the internal ordering.
This function sets the value z_{ij} for grid point (i,j) of the array za to z.
This function returns the value z_{ij} for grid point (i,j) stored in the array za.
This function returns the index corresponding to the grid point (i,j). The index is given by j*xsize + i.
Next: General Discrete Distributions, Previous: The Type-2 Gumbel Distribution, Up: Random Number Distributions [Index]
This function returns an array of K random variates from a Dirichlet distribution of order K-1. The distribution function is
p(\theta_1, ..., \theta_K) d\theta_1 ... d\theta_K =
(1/Z) \prod_{i=1}^K \theta_i^{\alpha_i - 1} \delta(1 -\sum_{i=1}^K \theta_i) d\theta_1 ... d\theta_K
for theta_i >= 0 and alpha_i > 0. The delta function ensures that \sum \theta_i = 1. The normalization factor Z is
Z = {\prod_{i=1}^K \Gamma(\alpha_i)} / {\Gamma( \sum_{i=1}^K \alpha_i)}
The random variates are generated by sampling K values from gamma distributions with parameters a=alpha_i, b=1, and renormalizing. See A.M. Law, W.D. Kelton, Simulation Modeling and Analysis (1991).
This function computes the probability density p(\theta_1, ... , \theta_K) at theta[K] for a Dirichlet distribution with parameters alpha[K], using the formula given above.
This function computes the logarithm of the probability density p(\theta_1, ... , \theta_K) for a Dirichlet distribution with parameters alpha[K].
Next: Examples with Simulated Annealing, Previous: Simulated Annealing algorithm, Up: Simulated Annealing [Index]
This function performs a simulated annealing search through a given space. The space is specified by providing the functions Ef and distance. The simulated annealing steps are generated using the random number generator r and the function take_step.
The starting configuration of the system should be given by x0_p.
The routine offers two modes for updating configurations, a fixed-size
mode and a variable-size mode. In the fixed-size mode the configuration
is stored as a single block of memory of size element_size.
Copies of this configuration are created, copied and destroyed
internally using the standard library functions malloc,
memcpy and free. The function pointers copyfunc,
copy_constructor and destructor should be null pointers in
fixed-size mode. In the variable-size mode the functions
copyfunc, copy_constructor and destructor are used to
create, copy and destroy configurations internally. The variable
element_size should be zero in the variable-size mode.
The params structure (described below) controls the run by providing the temperature schedule and other tunable parameters to the algorithm.
On exit the best result achieved during the search is placed in
*x0_p. If the annealing process has been successful this
should be a good approximation to the optimal point in the space.
If the function pointer print_position is not null, a debugging
log will be printed to stdout with the following columns:
#-iter #-evals temperature position energy best_energy
and the output of the function print_position itself. If print_position is null then no information is printed.
The simulated annealing routines require several user-specified functions to define the configuration space and energy function. The prototypes for these functions are given below.
This function type should return the energy of a configuration xp.
double (*gsl_siman_Efunc_t) (void *xp)
This function type should modify the configuration xp using a random step taken from the generator r, up to a maximum distance of step_size.
void (*gsl_siman_step_t) (const gsl_rng *r, void *xp,
double step_size)
This function type should return the distance between two configurations xp and yp.
double (*gsl_siman_metric_t) (void *xp, void *yp)
This function type should print the contents of the configuration xp.
void (*gsl_siman_print_t) (void *xp)
This function type should copy the configuration source into dest.
void (*gsl_siman_copy_t) (void *source, void *dest)
This function type should create a new copy of the configuration xp.
void * (*gsl_siman_copy_construct_t) (void *xp)
This function type should destroy the configuration xp, freeing its memory.
void (*gsl_siman_destroy_t) (void *xp)
These are the parameters that control a run of gsl_siman_solve.
This structure contains all the information needed to control the
search, beyond the energy function, the step function and the initial
guess.
int n_triesThe number of points to try for each step.
int iters_fixed_TThe number of iterations at each temperature.
double step_sizeThe maximum step size in the random walk.
double k, t_initial, mu_t, t_minThe parameters of the Boltzmann distribution and cooling schedule.
Next: Examples with Simulated Annealing, Previous: Simulated Annealing algorithm, Up: Simulated Annealing [Index]
Next: Level 3 CBLAS Functions, Previous: Level 1 CBLAS Functions, Up: GSL CBLAS Library [Index]
Next: Level 3 CBLAS Functions, Previous: Level 1 CBLAS Functions, Up: GSL CBLAS Library [Index]
Previous: Combination Examples, Up: Combinations [Index]
Further information on combinations can be found in,
Previous: DWT in one dimension, Up: DWT Transform Functions [Index]
The library provides functions to perform two-dimensional discrete wavelet transforms on square matrices. The matrix dimensions must be an integer power of two. There are two possible orderings of the rows and columns in the two-dimensional wavelet transform, referred to as the “standard” and “non-standard” forms.
The “standard” transform performs a complete discrete wavelet transform on the rows of the matrix, followed by a separate complete discrete wavelet transform on the columns of the resulting row-transformed matrix. This procedure uses the same ordering as a two-dimensional Fourier transform.
The “non-standard” transform is performed in interleaved passes on the rows and columns of the matrix for each level of the transform. The first level of the transform is applied to the matrix rows, and then to the matrix columns. This procedure is then repeated across the rows and columns of the data for the subsequent levels of the transform, until the full discrete wavelet transform is complete. The non-standard form of the discrete wavelet transform is typically used in image analysis.
The functions described in this section are declared in the header file gsl_wavelet2d.h.
These functions compute two-dimensional in-place forward and inverse
discrete wavelet transforms in standard form on the
array data stored in row-major form with dimensions size1
and size2 and physical row length tda. The dimensions must
be equal (square matrix) and are restricted to powers of two. For the
transform version of the function the argument dir can be
either forward (+1) or backward (-1). A
workspace work of the appropriate size must be provided. On exit,
the appropriate elements of the array data are replaced by their
two-dimensional wavelet transform.
The functions return a status of GSL_SUCCESS upon successful
completion. GSL_EINVAL is returned if size1 and
size2 are not equal and integer powers of 2, or if insufficient
workspace is provided.
These functions compute the two-dimensional in-place wavelet transform on a matrix a.
These functions compute the two-dimensional wavelet transform in non-standard form.
These functions compute the non-standard form of the two-dimensional in-place wavelet transform on a matrix a.
Previous: DWT in one dimension, Up: DWT Transform Functions [Index]
Next: Random number environment variables, Previous: Sampling from a random number generator, Up: Random Number Generation [Index]
The following functions provide information about an existing generator. You should use them in preference to hard-coding the generator parameters into your own code.
This function returns a pointer to the name of the generator. For example,
printf ("r is a '%s' generator\n",
gsl_rng_name (r));
would print something like r is a 'taus' generator.
gsl_rng_max returns the largest value that gsl_rng_get
can return.
gsl_rng_min returns the smallest value that gsl_rng_get
can return. Usually this value is zero. There are some generators with
algorithms that cannot return zero, and for these generators the minimum
value is 1.
These functions return a pointer to the state of generator r and its size. You can use this information to access the state directly. For example, the following code will write the state of a generator to a stream,
void * state = gsl_rng_state (r); size_t n = gsl_rng_size (r); fwrite (state, n, 1, stream);
This function returns a pointer to an array of all the available generator types, terminated by a null pointer. The function should be called once at the start of the program, if needed. The following code fragment shows how to iterate over the array of generator types to print the names of the available algorithms,
const gsl_rng_type **t, **t0;
t0 = gsl_rng_types_setup ();
printf ("Available generators:\n");
for (t = t0; *t != 0; t++)
{
printf ("%s\n", (*t)->name);
}
Next: Random number environment variables, Previous: Sampling from a random number generator, Up: Random Number Generation [Index]
Next: Copying random number generator state, Previous: Auxiliary random number generator functions, Up: Random Number Generation [Index]
The library allows you to choose a default generator and seed from the
environment variables GSL_RNG_TYPE and GSL_RNG_SEED and
the function gsl_rng_env_setup. This makes it easy try out
different generators and seeds without having to recompile your program.
This function reads the environment variables GSL_RNG_TYPE and
GSL_RNG_SEED and uses their values to set the corresponding
library variables gsl_rng_default and
gsl_rng_default_seed. These global variables are defined as
follows,
extern const gsl_rng_type *gsl_rng_default extern unsigned long int gsl_rng_default_seed
The environment variable GSL_RNG_TYPE should be the name of a
generator, such as taus or mt19937. The environment
variable GSL_RNG_SEED should contain the desired seed value. It
is converted to an unsigned long int using the C library function
strtoul.
If you don’t specify a generator for GSL_RNG_TYPE then
gsl_rng_mt19937 is used as the default. The initial value of
gsl_rng_default_seed is zero.
Here is a short program which shows how to create a global
generator using the environment variables GSL_RNG_TYPE and
GSL_RNG_SEED,
#include <stdio.h>
#include <gsl/gsl_rng.h>
gsl_rng * r; /* global generator */
int
main (void)
{
const gsl_rng_type * T;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
printf ("generator type: %s\n", gsl_rng_name (r));
printf ("seed = %lu\n", gsl_rng_default_seed);
printf ("first value = %lu\n", gsl_rng_get (r));
gsl_rng_free (r);
return 0;
}
Running the program without any environment variables uses the initial
defaults, an mt19937 generator with a seed of 0,
$ ./a.out
generator type: mt19937 seed = 0 first value = 4293858116
By setting the two variables on the command line we can change the default generator and the seed,
$ GSL_RNG_TYPE="taus" GSL_RNG_SEED=123 ./a.out GSL_RNG_TYPE=taus GSL_RNG_SEED=123 generator type: taus seed = 123 first value = 2720986350
Next: Copying random number generator state, Previous: Auxiliary random number generator functions, Up: Random Number Generation [Index]
Previous: BLAS Examples, Up: BLAS Support [Index]
Information on the BLAS standards, including both the legacy and updated interface standards, is available online from the BLAS Homepage and BLAS Technical Forum web-site.
The following papers contain the specifications for Level 1, Level 2 and Level 3 BLAS.
Postscript versions of the latter two papers are available from http://www.netlib.org/blas/. A CBLAS wrapper for Fortran BLAS libraries is available from the same location.
gsl-ref-html-2.3/Complete-Orthogonal-Decomposition.html 0000664 0001750 0001750 00000022340 13055414462 021322 0 ustar edd eddNext: Singular Value Decomposition, Previous: QR Decomposition with Column Pivoting, Up: Linear Algebra [Index]
The complete orthogonal decomposition of a M-by-N matrix A is a generalization of the QR decomposition with column pivoting, given by
A P = Q [ R11 0 ] Z
[ 0 0 ]
where P is a N-by-N permutation matrix, Q is M-by-M orthogonal, R_{11} is r-by-r upper triangular, with r = {\rm rank}(A), and Z is N-by-N orthogonal. If A has full rank, then R_{11} = R, Z = I and this reduces to the QR decomposition with column pivoting. The advantage of using the complete orthogonal decomposition for rank deficient matrices is the ability to compute the minimum norm solution to the linear least squares problem Ax = b, which is given by
x = P Z^T [ R11^-1 c1 ]
[ 0 ]
and the vector c_1 is the first r elements of Q^T b.
These functions factor the M-by-N matrix A into the decomposition A = Q R Z P^T. The rank of A
is computed as the number of diagonal elements of R greater than the tolerance tol and output in rank.
If tol is not specified, a default value is used (see gsl_linalg_QRPT_rank). On output, the permutation
matrix P is stored in p. The matrix R_{11} is stored in the upper rank-by-rank block of A.
The matrices Q and Z are encoded in packed storage in A on output. The vectors tau_Q and tau_Z
contain the Householder scalars corresponding to the matrices Q and Z respectively and must be
of length k = \min(M,N). The vector work is additional workspace of length N.
This function finds the least squares solution to the overdetermined
system A x = b where the matrix A has more rows than
columns. The least squares solution minimizes the Euclidean norm of the
residual, ||b - A x||. The routine requires as input
the QRZ decomposition of A into (QRZ, tau_Q, tau_Z, p, rank)
given by gsl_linalg_COD_decomp. The solution is returned in x. The
residual is computed as a by-product and stored in residual.
This function unpacks the encoded QRZ decomposition (QRZ, tau_Q, tau_Z, rank) into the matrices Q, R, and Z, where Q is M-by-M, R is M-by-N, and Z is N-by-N.
This function multiplies the input matrix A on the right by Z, A' = A Z using the encoded QRZ decomposition (QRZ, tau_Z, rank). A must have N columns but may have any number of rows. Additional workspace of length M is provided in work.
Next: Singular Value Decomposition, Previous: QR Decomposition with Column Pivoting, Up: Linear Algebra [Index]
Next: Reading and writing multisets, Previous: Multiset properties, Up: Multisets [Index]
This function advances the multiset c to the next multiset element in
lexicographic order and returns GSL_SUCCESS. If no further multisets
elements are available it returns GSL_FAILURE and leaves c
unmodified. Starting with the first multiset and repeatedly applying this
function will iterate through all possible multisets of a given order.
This function steps backwards from the multiset c to the previous
multiset element in lexicographic order, returning GSL_SUCCESS. If no
previous multiset is available it returns GSL_FAILURE and leaves c
unmodified.
Previous: Matrices, Up: Vectors and Matrices [Index]
The block, vector and matrix objects in GSL follow the valarray
model of C++. A description of this model can be found in the following
reference,
Next: Exponential Integrals, Previous: Error Functions, Up: Special Functions [Index]
The functions described in this section are declared in the header file gsl_sf_exp.h.
| • Exponential Function: | ||
| • Relative Exponential Functions: | ||
| • Exponentiation With Error Estimate: |
Next: Gegenbauer Functions, Previous: Fermi-Dirac Function, Up: Special Functions [Index]
This following routines compute the gamma and beta functions in their full and incomplete forms, as well as various kinds of factorials. The functions described in this section are declared in the header file gsl_sf_gamma.h.
| • Gamma Functions: | ||
| • Factorials: | ||
| • Pochhammer Symbol: | ||
| • Incomplete Gamma Functions: | ||
| • Beta Functions: | ||
| • Incomplete Beta Function: |
Next: Least-Squares Fitting, Previous: Multidimensional Root-Finding, Up: Top [Index]
This chapter describes routines for finding minima of arbitrary multidimensional functions. The library provides low level components for a variety of iterative minimizers and convergence tests. These can be combined by the user to achieve the desired solution, while providing full access to the intermediate steps of the algorithms. Each class of methods uses the same framework, so that you can switch between minimizers at runtime without needing to recompile your program. Each instance of a minimizer keeps track of its own state, allowing the minimizers to be used in multi-threaded programs. The minimization algorithms can be used to maximize a function by inverting its sign.
The header file gsl_multimin.h contains prototypes for the minimization functions and related declarations.
Next: Integrands with singular weight functions, Previous: Integrands without weight functions, Up: Numerical Integration Introduction [Index]
For integrands with weight functions the algorithms use Clenshaw-Curtis quadrature rules.
A Clenshaw-Curtis rule begins with an n-th order Chebyshev polynomial approximation to the integrand. This polynomial can be integrated exactly to give an approximation to the integral of the original function. The Chebyshev expansion can be extended to higher orders to improve the approximation and provide an estimate of the error.
gsl-ref-html-2.3/Lambert-W-Functions.html 0000664 0001750 0001750 00000012170 13055414531 016363 0 ustar edd eddNext: Legendre Functions and Spherical Harmonics, Previous: Laguerre Functions, Up: Special Functions [Index]
Lambert’s W functions, W(x), are defined to be solutions of the equation W(x) \exp(W(x)) = x. This function has multiple branches for x < 0; however, it has only two real-valued branches. We define W_0(x) to be the principal branch, where W > -1 for x < 0, and W_{-1}(x) to be the other real branch, where W < -1 for x < 0. The Lambert functions are declared in the header file gsl_sf_lambert.h.
These compute the principal branch of the Lambert W function, W_0(x).
These compute the secondary real-valued branch of the Lambert W function, W_{-1}(x).
Next: Running Statistics, Previous: Random Number Distributions, Up: Top [Index]
This chapter describes the statistical functions in the library. The basic statistical functions include routines to compute the mean, variance and standard deviation. More advanced functions allow you to calculate absolute deviations, skewness, and kurtosis as well as the median and arbitrary percentiles. The algorithms use recurrence relations to compute average quantities in a stable way, without large intermediate values that might overflow.
The functions are available in versions for datasets in the standard
floating-point and integer types. The versions for double precision
floating-point data have the prefix gsl_stats and are declared in
the header file gsl_statistics_double.h. The versions for integer
data have the prefix gsl_stats_int and are declared in the header
file gsl_statistics_int.h. All the functions operate on C
arrays with a stride parameter specifying the spacing between
elements.
Next: Running Statistics, Previous: Random Number Distributions, Up: Top [Index]
Next: Sparse Iterative Solvers, Up: Sparse Linear Algebra [Index]
This chapter is primarily concerned with the solution of the linear system
A x = b
where A is a general square n-by-n non-singular sparse matrix, x is an unknown n-by-1 vector, and b is a given n-by-1 right hand side vector. There exist many methods for solving such sparse linear systems, which broadly fall into either direct or iterative categories. Direct methods include LU and QR decompositions, while iterative methods start with an initial guess for the vector x and update the guess through iteration until convergence. GSL does not currently provide any direct sparse solvers.
gsl-ref-html-2.3/Matrix-allocation.html 0000664 0001750 0001750 00000012327 13055414466 016225 0 ustar edd eddNext: Accessing matrix elements, Up: Matrices [Index]
The functions for allocating memory to a matrix follow the style of
malloc and free. They also perform their own error
checking. If there is insufficient memory available to allocate a matrix
then the functions call the GSL error handler (with an error number of
GSL_ENOMEM) in addition to returning a null pointer. Thus if you
use the library error handler to abort your program then it isn’t
necessary to check every alloc.
This function creates a matrix of size n1 rows by n2 columns, returning a pointer to a newly initialized matrix struct. A new block is allocated for the elements of the matrix, and stored in the block component of the matrix struct. The block is “owned” by the matrix, and will be deallocated when the matrix is deallocated.
This function allocates memory for a matrix of size n1 rows by n2 columns and initializes all the elements of the matrix to zero.
This function frees a previously allocated matrix m. If the
matrix was created using gsl_matrix_alloc then the block
underlying the matrix will also be deallocated. If the matrix has
been created from another object then the memory is still owned by
that object and will not be deallocated.
Next: Testing the Sign of Numbers, Previous: Elementary Functions, Up: Mathematical Functions [Index]
A common complaint about the standard C library is its lack of a function for calculating (small) integer powers. GSL provides some simple functions to fill this gap. For reasons of efficiency, these functions do not check for overflow or underflow conditions.
These routines computes the power x^n for integer n. The
power is computed efficiently—for example, x^8 is computed as
((x^2)^2)^2, requiring only 3 multiplications. A version of this
function which also computes the numerical error in the result is
available as gsl_sf_pow_int_e.
These functions can be used to compute small integer powers x^2,
x^3, etc. efficiently. The functions will be inlined when
HAVE_INLINE is defined, so that use of these functions
should be as efficient as explicitly writing the corresponding
product expression.
#include <gsl/gsl_math.h> double y = gsl_pow_4 (3.141) /* compute 3.141**4 */
Next: Linear regression, Up: Least-Squares Fitting [Index]
Least-squares fits are found by minimizing \chi^2 (chi-squared), the weighted sum of squared residuals over n experimental datapoints (x_i, y_i) for the model Y(c,x),
\chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2
The p parameters of the model are c = {c_0, c_1, …}. The weight factors w_i are given by w_i = 1/\sigma_i^2, where \sigma_i is the experimental error on the data-point y_i. The errors are assumed to be Gaussian and uncorrelated. For unweighted data the chi-squared sum is computed without any weight factors.
The fitting routines return the best-fit parameters c and their p \times p covariance matrix. The covariance matrix measures the statistical errors on the best-fit parameters resulting from the errors on the data, \sigma_i, and is defined as C_{ab} = <\delta c_a \delta c_b> where < > denotes an average over the Gaussian error distributions of the underlying datapoints.
The covariance matrix is calculated by error propagation from the data errors \sigma_i. The change in a fitted parameter \delta c_a caused by a small change in the data \delta y_i is given by
\delta c_a = \sum_i (dc_a/dy_i) \delta y_i
allowing the covariance matrix to be written in terms of the errors on the data,
C_{ab} = \sum_{i,j} (dc_a/dy_i) (dc_b/dy_j) <\delta y_i \delta y_j>
For uncorrelated data the fluctuations of the underlying datapoints satisfy <\delta y_i \delta y_j> = \sigma_i^2 \delta_{ij}, giving a corresponding parameter covariance matrix of
C_{ab} = \sum_i (1/w_i) (dc_a/dy_i) (dc_b/dy_i)
When computing the covariance matrix for unweighted data, i.e. data with unknown errors, the weight factors w_i in this sum are replaced by the single estimate w = 1/\sigma^2, where \sigma^2 is the computed variance of the residuals about the best-fit model, \sigma^2 = \sum (y_i - Y(c,x_i))^2 / (n-p). This is referred to as the variance-covariance matrix.
The standard deviations of the best-fit parameters are given by the square root of the corresponding diagonal elements of the covariance matrix, \sigma_{c_a} = \sqrt{C_{aa}}. The correlation coefficient of the fit parameters c_a and c_b is given by \rho_{ab} = C_{ab} / \sqrt{C_{aa} C_{bb}}.
Next: Linear regression, Up: Least-Squares Fitting [Index]
Next: The t-distribution, Previous: The Chi-squared Distribution, Up: Random Number Distributions [Index]
The F-distribution arises in statistics. If Y_1 and Y_2 are chi-squared deviates with \nu_1 and \nu_2 degrees of freedom then the ratio,
X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }
has an F-distribution F(x;\nu_1,\nu_2).
This function returns a random variate from the F-distribution with degrees of freedom nu1 and nu2. The distribution function is,
p(x) dx =
{ \Gamma((\nu_1 + \nu_2)/2)
\over \Gamma(\nu_1/2) \Gamma(\nu_2/2) }
\nu_1^{\nu_1/2} \nu_2^{\nu_2/2}
x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}
for x >= 0.
This function computes the probability density p(x) at x for an F-distribution with nu1 and nu2 degrees of freedom, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the F-distribution with nu1 and nu2 degrees of freedom.
Next: Lambert W Functions, Previous: Hypergeometric Functions, Up: Special Functions [Index]
The generalized Laguerre polynomials are defined in terms of confluent hypergeometric functions as L^a_n(x) = ((a+1)_n / n!) 1F1(-n,a+1,x), and are sometimes referred to as the associated Laguerre polynomials. They are related to the plain Laguerre polynomials L_n(x) by L^0_n(x) = L_n(x) and L^k_n(x) = (-1)^k (d^k/dx^k) L_(n+k)(x). For more information see Abramowitz & Stegun, Chapter 22.
The functions described in this section are declared in the header file gsl_sf_laguerre.h.
These routines evaluate the generalized Laguerre polynomials L^a_1(x), L^a_2(x), L^a_3(x) using explicit representations.
These routines evaluate the generalized Laguerre polynomials L^a_n(x) for a > -1, n >= 0.
Next: Copying Histograms, Previous: The histogram struct, Up: Histograms [Index]
The functions for allocating memory to a histogram follow the style of
malloc and free. In addition they also perform their own
error checking. If there is insufficient memory available to allocate a
histogram then the functions call the error handler (with an error
number of GSL_ENOMEM) in addition to returning a null pointer.
Thus if you use the library error handler to abort your program then it
isn’t necessary to check every histogram alloc.
This function allocates memory for a histogram with n bins, and
returns a pointer to a newly created gsl_histogram struct. If
insufficient memory is available a null pointer is returned and the
error handler is invoked with an error code of GSL_ENOMEM. The
bins and ranges are not initialized, and should be prepared using one of
the range-setting functions below in order to make the histogram ready
for use.
This function sets the ranges of the existing histogram h using
the array range of size size. The values of the histogram
bins are reset to zero. The range array should contain the
desired bin limits. The ranges can be arbitrary, subject to the
restriction that they are monotonically increasing.
The following example shows how to create a histogram with logarithmic bins with ranges [1,10), [10,100) and [100,1000).
gsl_histogram * h = gsl_histogram_alloc (3);
/* bin[0] covers the range 1 <= x < 10 */
/* bin[1] covers the range 10 <= x < 100 */
/* bin[2] covers the range 100 <= x < 1000 */
double range[4] = { 1.0, 10.0, 100.0, 1000.0 };
gsl_histogram_set_ranges (h, range, 4);
Note that the size of the range array should be defined to be one element bigger than the number of bins. The additional element is required for the upper value of the final bin.
This function sets the ranges of the existing histogram h to cover the range xmin to xmax uniformly. The values of the histogram bins are reset to zero. The bin ranges are shown in the table below,
bin[0] corresponds to xmin <= x < xmin + d bin[1] corresponds to xmin + d <= x < xmin + 2 d ...... bin[n-1] corresponds to xmin + (n-1)d <= x < xmax
where d is the bin spacing, d = (xmax-xmin)/n.
This function frees the histogram h and all of the memory associated with it.
Next: Copying Histograms, Previous: The histogram struct, Up: Histograms [Index]
Next: GSL CBLAS Library, Previous: Contributors to GSL, Up: Top [Index]
For applications using autoconf the standard macro
AC_CHECK_LIB can be used to link with GSL automatically
from a configure script. The library itself depends on the
presence of a CBLAS and math library as well, so these must also be
located before linking with the main libgsl file. The following
commands should be placed in the configure.ac file to perform
these tests,
AC_CHECK_LIB([m],[cos]) AC_CHECK_LIB([gslcblas],[cblas_dgemm]) AC_CHECK_LIB([gsl],[gsl_blas_dgemm])
It is important to check for libm and libgslcblas before
libgsl, otherwise the tests will fail. Assuming the libraries
are found the output during the configure stage looks like this,
checking for cos in -lm... yes checking for cblas_dgemm in -lgslcblas... yes checking for gsl_blas_dgemm in -lgsl... yes
If the library is found then the tests will define the macros
HAVE_LIBGSL, HAVE_LIBGSLCBLAS, HAVE_LIBM and add
the options -lgsl -lgslcblas -lm to the variable LIBS.
The tests above will find any version of the library. They are suitable for general use, where the versions of the functions are not important. An alternative macro is available in the file gsl.m4 to test for a specific version of the library. To use this macro simply add the following line to your configure.in file instead of the tests above:
AX_PATH_GSL(GSL_VERSION,
[action-if-found],
[action-if-not-found])
The argument GSL_VERSION should be the two or three digit
MAJOR.MINOR or MAJOR.MINOR.MICRO version number of the release
you require. A suitable choice for action-if-not-found is,
AC_MSG_ERROR(could not find required version of GSL)
Then you can add the variables GSL_LIBS and GSL_CFLAGS to
your Makefile.am files to obtain the correct compiler flags.
GSL_LIBS is equal to the output of the gsl-config --libs
command and GSL_CFLAGS is equal to gsl-config --cflags
command. For example,
libfoo_la_LDFLAGS = -lfoo $(GSL_LIBS) -lgslcblas
Note that the macro AX_PATH_GSL needs to use the C compiler so it
should appear in the configure.in file before the macro
AC_LANG_CPLUSPLUS for programs that use C++.
To test for inline the following test should be placed in your
configure.in file,
AC_C_INLINE if test "$ac_cv_c_inline" != no ; then AC_DEFINE(HAVE_INLINE,1) AC_SUBST(HAVE_INLINE) fi
and the macro will then be defined in the compilation flags or by including the file config.h before any library headers.
The following autoconf test will check for extern inline,
dnl Check for "extern inline", using a modified version
dnl of the test for AC_C_INLINE from acspecific.mt
dnl
AC_CACHE_CHECK([for extern inline], ac_cv_c_extern_inline,
[ac_cv_c_extern_inline=no
AC_TRY_COMPILE([extern $ac_cv_c_inline double foo(double x);
extern $ac_cv_c_inline double foo(double x) { return x+1.0; };
double foo (double x) { return x + 1.0; };],
[ foo(1.0) ],
[ac_cv_c_extern_inline="yes"])
])
if test "$ac_cv_c_extern_inline" != no ; then
AC_DEFINE(HAVE_INLINE,1)
AC_SUBST(HAVE_INLINE)
fi
The substitution of portability functions can be made automatically if
you use autoconf. For example, to test whether the BSD function
hypot is available you can include the following line in the
configure file configure.in for your application,
AC_CHECK_FUNCS(hypot)
and place the following macro definitions in the file config.h.in,
/* Substitute gsl_hypot for missing system hypot */ #ifndef HAVE_HYPOT #define hypot gsl_hypot #endif
The application source files can then use the include command
#include <config.h> to substitute gsl_hypot for each
occurrence of hypot when hypot is not available.
Next: GSL CBLAS Library, Previous: Contributors to GSL, Up: Top [Index]
Next: Accessing vector elements, Up: Vectors [Index]
The functions for allocating memory to a vector follow the style of
malloc and free. In addition they also perform their own
error checking. If there is insufficient memory available to allocate a
vector then the functions call the GSL error handler (with an error
number of GSL_ENOMEM) in addition to returning a null
pointer. Thus if you use the library error handler to abort your program
then it isn’t necessary to check every alloc.
This function creates a vector of length n, returning a pointer to a newly initialized vector struct. A new block is allocated for the elements of the vector, and stored in the block component of the vector struct. The block is “owned” by the vector, and will be deallocated when the vector is deallocated.
This function allocates memory for a vector of length n and initializes all the elements of the vector to zero.
This function frees a previously allocated vector v. If the
vector was created using gsl_vector_alloc then the block
underlying the vector will also be deallocated. If the vector has
been created from another object then the memory is still owned by
that object and will not be deallocated.
Next: Example statistical programs, Previous: Maximum and Minimum values, Up: Statistics [Index]
The median and percentile functions described in this section operate on sorted data. For convenience we use quantiles, measured on a scale of 0 to 1, instead of percentiles (which use a scale of 0 to 100).
This function returns the median value of sorted_data, a dataset
of length n with stride stride. The elements of the array
must be in ascending numerical order. There are no checks to see
whether the data are sorted, so the function gsl_sort should
always be used first.
When the dataset has an odd number of elements the median is the value of element (n-1)/2. When the dataset has an even number of elements the median is the mean of the two nearest middle values, elements (n-1)/2 and n/2. Since the algorithm for computing the median involves interpolation this function always returns a floating-point number, even for integer data types.
This function returns a quantile value of sorted_data, a double-precision array of length n with stride stride. The elements of the array must be in ascending numerical order. The quantile is determined by the f, a fraction between 0 and 1. For example, to compute the value of the 75th percentile f should have the value 0.75.
There are no checks to see whether the data are sorted, so the function
gsl_sort should always be used first.
The quantile is found by interpolation, using the formula
quantile = (1 - \delta) x_i + \delta x_{i+1}
where i is floor((n - 1)f) and \delta is
(n-1)f - i.
Thus the minimum value of the array (data[0*stride]) is given by
f equal to zero, the maximum value (data[(n-1)*stride]) is
given by f equal to one and the median value is given by f
equal to 0.5. Since the algorithm for computing quantiles involves
interpolation this function always returns a floating-point number, even
for integer data types.
Next: Example statistical programs, Previous: Maximum and Minimum values, Up: Statistics [Index]
Next: Minimization Iteration, Previous: Initializing the Minimizer, Up: One dimensional Minimization [Index]
You must provide a continuous function of one variable for the
minimizers to operate on. In order to allow for general parameters the
functions are defined by a gsl_function data type
(see Providing the function to solve).
Next: Coulomb Functions, Previous: Bessel Functions, Up: Special Functions [Index]
The Clausen function is defined by the following integral,
Cl_2(x) = - \int_0^x dt \log(2 \sin(t/2))
It is related to the dilogarithm by Cl_2(\theta) = \Im Li_2(\exp(i\theta)). The Clausen functions are declared in the header file gsl_sf_clausen.h.
These routines compute the Clausen integral Cl_2(x).
Next: Polygamma Function, Previous: Digamma Function, Up: Psi (Digamma) Function [Index]
These routines compute the Trigamma function \psi'(n) for positive integer n.
These routines compute the Trigamma function \psi'(x) for general x.
Next: Measurement of Time, Previous: Astronomy and Astrophysics, Up: Physical Constants [Index]
GSL_CONST_MKSA_ELECTRON_CHARGEThe charge of the electron, e.
GSL_CONST_MKSA_ELECTRON_VOLTThe energy of 1 electron volt, eV.
GSL_CONST_MKSA_UNIFIED_ATOMIC_MASSThe unified atomic mass, amu.
GSL_CONST_MKSA_MASS_ELECTRONThe mass of the electron, m_e.
GSL_CONST_MKSA_MASS_MUONThe mass of the muon, m_\mu.
GSL_CONST_MKSA_MASS_PROTONThe mass of the proton, m_p.
GSL_CONST_MKSA_MASS_NEUTRONThe mass of the neutron, m_n.
GSL_CONST_NUM_FINE_STRUCTUREThe electromagnetic fine structure constant \alpha.
GSL_CONST_MKSA_RYDBERGThe Rydberg constant, Ry, in units of energy. This is related to the Rydberg inverse wavelength R_\infty by Ry = h c R_\infty.
GSL_CONST_MKSA_BOHR_RADIUSThe Bohr radius, a_0.
GSL_CONST_MKSA_ANGSTROMThe length of 1 angstrom.
GSL_CONST_MKSA_BARNThe area of 1 barn.
GSL_CONST_MKSA_BOHR_MAGNETONThe Bohr Magneton, \mu_B.
GSL_CONST_MKSA_NUCLEAR_MAGNETONThe Nuclear Magneton, \mu_N.
GSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENTThe absolute value of the magnetic moment of the electron, \mu_e. The physical magnetic moment of the electron is negative.
GSL_CONST_MKSA_PROTON_MAGNETIC_MOMENTThe magnetic moment of the proton, \mu_p.
GSL_CONST_MKSA_THOMSON_CROSS_SECTIONThe Thomson cross section, \sigma_T.
GSL_CONST_MKSA_DEBYEThe electric dipole moment of 1 Debye, D.
Next: Trigonometric Integrals, Previous: Hyperbolic Integrals, Up: Exponential Integrals [Index]
These routines compute the third-order exponential integral Ei_3(x) = \int_0^xdt \exp(-t^3) for x >= 0.
Next: Nonlinear Least-Squares TRS Double Dogleg, Previous: Nonlinear Least-Squares TRS Levenberg-Marquardt with Geodesic Acceleration, Up: Nonlinear Least-Squares TRS Overview [Index]
This is Powell’s dogleg method, which finds an approximate solution to the trust region subproblem, by restricting its search to a piecewise linear “dogleg” path, composed of the origin, the Cauchy point which represents the model minimizer along the steepest descent direction, and the Gauss-Newton point, which is the overall minimizer of the unconstrained model. The Gauss-Newton step is calculated by solving
J_k \delta_gn = -f_k
which is the main computational task for each iteration, but only needs to be performed once per iteration. If the Gauss-Newton point is inside the trust region, it is selected as the step. If it is outside, the method then calculates the Cauchy point, which is located along the gradient direction. If the Cauchy point is also outside the trust region, the method assumes that it is still far from the minimum and so proceeds along the gradient direction, truncating the step at the trust region boundary. If the Cauchy point is inside the trust region, with the Gauss-Newton point outside, the method uses a dogleg step, which is a linear combination of the gradient direction and the Gauss-Newton direction, stopping at the trust region boundary.
gsl-ref-html-2.3/The-Bernoulli-Distribution.html 0000664 0001750 0001750 00000011077 13055414506 017760 0 ustar edd eddNext: The Binomial Distribution, Previous: The Poisson Distribution, Up: Random Number Distributions [Index]
This function returns either 0 or 1, the result of a Bernoulli trial with probability p. The probability distribution for a Bernoulli trial is,
p(0) = 1 - p p(1) = p
This function computes the probability p(k) of obtaining k from a Bernoulli distribution with probability parameter p, using the formula given above.
Next: Probability functions, Previous: Complementary Error Function, Up: Error Functions [Index]
These routines compute the logarithm of the complementary error function \log(\erfc(x)).
Next: Error Handling, Previous: Introduction, Up: Top [Index]
This chapter describes how to compile programs that use GSL, and introduces its conventions.
Next: Overview of real data FFTs, Previous: Radix-2 FFT routines for complex data, Up: Fast Fourier Transforms [Index]
This section describes mixed-radix FFT algorithms for complex data. The mixed-radix functions work for FFTs of any length. They are a reimplementation of Paul Swarztrauber’s Fortran FFTPACK library. The theory is explained in the review article Self-sorting Mixed-radix FFTs by Clive Temperton. The routines here use the same indexing scheme and basic algorithms as FFTPACK.
The mixed-radix algorithm is based on sub-transform modules—highly optimized small length FFTs which are combined to create larger FFTs. There are efficient modules for factors of 2, 3, 4, 5, 6 and 7. The modules for the composite factors of 4 and 6 are faster than combining the modules for 2*2 and 2*3.
For factors which are not implemented as modules there is a fall-back to a general length-n module which uses Singleton’s method for efficiently computing a DFT. This module is O(n^2), and slower than a dedicated module would be but works for any length n. Of course, lengths which use the general length-n module will still be factorized as much as possible. For example, a length of 143 will be factorized into 11*13. Large prime factors are the worst case scenario, e.g. as found in n=2*3*99991, and should be avoided because their O(n^2) scaling will dominate the run-time (consult the document GSL FFT Algorithms included in the GSL distribution if you encounter this problem).
The mixed-radix initialization function gsl_fft_complex_wavetable_alloc
returns the list of factors chosen by the library for a given length
n. It can be used to check how well the length has been
factorized, and estimate the run-time. To a first approximation the
run-time scales as n \sum f_i, where the f_i are the
factors of n. For programs under user control you may wish to
issue a warning that the transform will be slow when the length is
poorly factorized. If you frequently encounter data lengths which
cannot be factorized using the existing small-prime modules consult
GSL FFT Algorithms for details on adding support for other
factors.
All the functions described in this section are declared in the header file gsl_fft_complex.h.
This function prepares a trigonometric lookup table for a complex FFT of
length n. The function returns a pointer to the newly allocated
gsl_fft_complex_wavetable if no errors were detected, and a null
pointer in the case of error. The length n is factorized into a
product of subtransforms, and the factors and their trigonometric
coefficients are stored in the wavetable. The trigonometric coefficients
are computed using direct calls to sin and cos, for
accuracy. Recursion relations could be used to compute the lookup table
faster, but if an application performs many FFTs of the same length then
this computation is a one-off overhead which does not affect the final
throughput.
The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified by calls to any of the other FFT functions. The same wavetable can be used for both forward and backward (or inverse) transforms of a given length.
This function frees the memory associated with the wavetable wavetable. The wavetable can be freed if no further FFTs of the same length will be needed.
These functions operate on a gsl_fft_complex_wavetable structure
which contains internal parameters for the FFT. It is not necessary to
set any of the components directly but it can sometimes be useful to
examine them. For example, the chosen factorization of the FFT length
is given and can be used to provide an estimate of the run-time or
numerical error. The wavetable structure is declared in the header file
gsl_fft_complex.h.
This is a structure that holds the factorization and trigonometric lookup tables for the mixed radix fft algorithm. It has the following components:
size_t nThis is the number of complex data points
size_t nfThis is the number of factors that the length n was decomposed into.
size_t factor[64]This is the array of factors. Only the first nf elements are
used.
gsl_complex * trigThis is a pointer to a preallocated trigonometric lookup table of
n complex elements.
gsl_complex * twiddle[64]This is an array of pointers into trig, giving the twiddle
factors for each pass.
The mixed radix algorithms require additional working space to hold the intermediate steps of the transform.
This function allocates a workspace for a complex transform of length n.
This function frees the memory associated with the workspace workspace. The workspace can be freed if no further FFTs of the same length will be needed.
The following functions compute the transform,
These functions compute forward, backward and inverse FFTs of length
n with stride stride, on the packed complex array
data, using a mixed radix decimation-in-frequency algorithm.
There is no restriction on the length n. Efficient modules are
provided for subtransforms of length 2, 3, 4, 5, 6 and 7. Any remaining
factors are computed with a slow, O(n^2), general-n
module. The caller must supply a wavetable containing the
trigonometric lookup tables and a workspace work. For the
transform version of the function the sign argument can be
either forward (-1) or backward (+1).
The functions return a value of 0 if no errors were detected. The
following gsl_errno conditions are defined for these functions:
GSL_EDOMThe length of the data n is not a positive integer (i.e. n is zero).
GSL_EINVALThe length of the data n and the length used to compute the given wavetable do not match.
Here is an example program which computes the FFT of a short pulse in a sample of length 630 (=2*3*3*5*7) using the mixed-radix algorithm.
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>
#define REAL(z,i) ((z)[2*(i)])
#define IMAG(z,i) ((z)[2*(i)+1])
int
main (void)
{
int i;
const int n = 630;
double data[2*n];
gsl_fft_complex_wavetable * wavetable;
gsl_fft_complex_workspace * workspace;
for (i = 0; i < n; i++)
{
REAL(data,i) = 0.0;
IMAG(data,i) = 0.0;
}
data[0] = 1.0;
for (i = 1; i <= 10; i++)
{
REAL(data,i) = REAL(data,n-i) = 1.0;
}
for (i = 0; i < n; i++)
{
printf ("%d: %e %e\n", i, REAL(data,i),
IMAG(data,i));
}
printf ("\n");
wavetable = gsl_fft_complex_wavetable_alloc (n);
workspace = gsl_fft_complex_workspace_alloc (n);
for (i = 0; i < (int) wavetable->nf; i++)
{
printf ("# factor %d: %zu\n", i,
wavetable->factor[i]);
}
gsl_fft_complex_forward (data, 1, n,
wavetable, workspace);
for (i = 0; i < n; i++)
{
printf ("%d: %e %e\n", i, REAL(data,i),
IMAG(data,i));
}
gsl_fft_complex_wavetable_free (wavetable);
gsl_fft_complex_workspace_free (workspace);
return 0;
}
Note that we have assumed that the program is using the default
gsl error handler (which calls abort for any errors). If
you are not using a safe error handler you would need to check the
return status of all the gsl routines.
Next: Overview of real data FFTs, Previous: Radix-2 FFT routines for complex data, Up: Fast Fourier Transforms [Index]
Next: Unix random number generators, Previous: Reading and writing random number generator state, Up: Random Number Generation [Index]
The functions described above make no reference to the actual algorithm used. This is deliberate so that you can switch algorithms without having to change any of your application source code. The library provides a large number of generators of different types, including simulation quality generators, generators provided for compatibility with other libraries and historical generators from the past.
The following generators are recommended for use in simulation. They have extremely long periods, low correlation and pass most statistical tests. For the most reliable source of uncorrelated numbers, the second-generation RANLUX generators have the strongest proof of randomness.
The MT19937 generator of Makoto Matsumoto and Takuji Nishimura is a
variant of the twisted generalized feedback shift-register algorithm,
and is known as the “Mersenne Twister” generator. It has a Mersenne
prime period of
2^19937 - 1 (about
10^6000) and is
equi-distributed in 623 dimensions. It has passed the DIEHARD
statistical tests. It uses 624 words of state per generator and is
comparable in speed to the other generators. The original generator used
a default seed of 4357 and choosing s equal to zero in
gsl_rng_set reproduces this. Later versions switched to 5489
as the default seed, you can choose this explicitly via gsl_rng_set
instead if you require it.
For more information see,
The generator gsl_rng_mt19937 uses the second revision of the
seeding procedure published by the two authors above in 2002. The
original seeding procedures could cause spurious artifacts for some seed
values. They are still available through the alternative generators
gsl_rng_mt19937_1999 and gsl_rng_mt19937_1998.
The generator ranlxs0 is a second-generation version of the
RANLUX algorithm of Lüscher, which produces “luxury random
numbers”. This generator provides single precision output (24 bits) at
three luxury levels ranlxs0, ranlxs1 and ranlxs2,
in increasing order of strength.
It uses double-precision floating point arithmetic internally and can be
significantly faster than the integer version of ranlux,
particularly on 64-bit architectures. The period of the generator is
about 10^171. The algorithm has mathematically proven properties and
can provide truly decorrelated numbers at a known level of randomness.
The higher luxury levels provide increased decorrelation between samples
as an additional safety margin.
Note that the range of allowed seeds for this generator is [0,2^31-1]. Higher seed values are wrapped modulo 2^31.
These generators produce double precision output (48 bits) from the
RANLXS generator. The library provides two luxury levels
ranlxd1 and ranlxd2, in increasing order of strength.
The ranlux generator is an implementation of the original
algorithm developed by Lüscher. It uses a
lagged-fibonacci-with-skipping algorithm to produce “luxury random
numbers”. It is a 24-bit generator, originally designed for
single-precision IEEE floating point numbers. This implementation is
based on integer arithmetic, while the second-generation versions
RANLXS and RANLXD described above provide floating-point
implementations which will be faster on many platforms.
The period of the generator is about 10^171. The algorithm has mathematically proven properties and
it can provide truly decorrelated numbers at a known level of
randomness. The default level of decorrelation recommended by Lüscher
is provided by gsl_rng_ranlux, while gsl_rng_ranlux389
gives the highest level of randomness, with all 24 bits decorrelated.
Both types of generator use 24 words of state per generator.
For more information see,
This is a combined multiple recursive generator by L’Ecuyer. Its sequence is,
z_n = (x_n - y_n) mod m_1
where the two underlying generators x_n and y_n are,
x_n = (a_1 x_{n-1} + a_2 x_{n-2} + a_3 x_{n-3}) mod m_1
y_n = (b_1 y_{n-1} + b_2 y_{n-2} + b_3 y_{n-3}) mod m_2
with coefficients a_1 = 0, a_2 = 63308, a_3 = -183326, b_1 = 86098, b_2 = 0, b_3 = -539608, and moduli m_1 = 2^31 - 1 = 2147483647 and m_2 = 2145483479.
The period of this generator is lcm(m_1^3-1, m_2^3-1), which is approximately 2^185 (about 10^56). It uses 6 words of state per generator. For more information see,
This is a fifth-order multiple recursive generator by L’Ecuyer, Blouin and Coutre. Its sequence is,
x_n = (a_1 x_{n-1} + a_5 x_{n-5}) mod m
with a_1 = 107374182, a_2 = a_3 = a_4 = 0, a_5 = 104480 and m = 2^31 - 1.
The period of this generator is about 10^46. It uses 5 words of state per generator. More information can be found in the following paper,
This is a maximally equidistributed combined Tausworthe generator by L’Ecuyer. The sequence is,
x_n = (s1_n ^^ s2_n ^^ s3_n)
where,
s1_{n+1} = (((s1_n&4294967294)<<12)^^(((s1_n<<13)^^s1_n)>>19))
s2_{n+1} = (((s2_n&4294967288)<< 4)^^(((s2_n<< 2)^^s2_n)>>25))
s3_{n+1} = (((s3_n&4294967280)<<17)^^(((s3_n<< 3)^^s3_n)>>11))
computed modulo
2^32. In the formulas above
^^
denotes “exclusive-or”. Note that the algorithm relies on the properties
of 32-bit unsigned integers and has been implemented using a bitmask
of 0xFFFFFFFF to make it work on 64 bit machines.
The period of this generator is 2^88 (about 10^26). It uses 3 words of state per generator. For more information see,
The generator gsl_rng_taus2 uses the same algorithm as
gsl_rng_taus but with an improved seeding procedure described in
the paper,
The generator gsl_rng_taus2 should now be used in preference to
gsl_rng_taus.
The gfsr4 generator is like a lagged-fibonacci generator, and
produces each number as an xor’d sum of four previous values.
r_n = r_{n-A} ^^ r_{n-B} ^^ r_{n-C} ^^ r_{n-D}
Ziff (ref below) notes that “it is now widely known” that two-tap registers (such as R250, which is described below) have serious flaws, the most obvious one being the three-point correlation that comes from the definition of the generator. Nice mathematical properties can be derived for GFSR’s, and numerics bears out the claim that 4-tap GFSR’s with appropriately chosen offsets are as random as can be measured, using the author’s test.
This implementation uses the values suggested the example on p392 of Ziff’s article: A=471, B=1586, C=6988, D=9689.
If the offsets are appropriately chosen (such as the one ones in this
implementation), then the sequence is said to be maximal; that means
that the period is 2^D - 1, where D is the longest lag.
(It is one less than 2^D because it is not permitted to have all
zeros in the ra[] array.) For this implementation with
D=9689 that works out to about 10^2917.
Note that the implementation of this generator using a 32-bit integer amounts to 32 parallel implementations of one-bit generators. One consequence of this is that the period of this 32-bit generator is the same as for the one-bit generator. Moreover, this independence means that all 32-bit patterns are equally likely, and in particular that 0 is an allowed random value. (We are grateful to Heiko Bauke for clarifying for us these properties of GFSR random number generators.)
For more information see,
Next: Unix random number generators, Previous: Reading and writing random number generator state, Up: Random Number Generation [Index]
Next: 2D Histogram allocation, Previous: Two dimensional histograms, Up: Histograms [Index]
Two dimensional histograms are defined by the following struct,
size_t nx, nyThis is the number of histogram bins in the x and y directions.
double * xrangeThe ranges of the bins in the x-direction are stored in an array of nx + 1 elements pointed to by xrange.
double * yrangeThe ranges of the bins in the y-direction are stored in an array of ny + 1 elements pointed to by yrange.
double * binThe counts for each bin are stored in an array pointed to by bin.
The bins are floating-point numbers, so you can increment them by
non-integer values if necessary. The array bin stores the two
dimensional array of bins in a single block of memory according to the
mapping bin(i,j) = bin[i * ny + j].
The range for bin(i,j) is given by xrange[i] to
xrange[i+1] in the x-direction and yrange[j] to
yrange[j+1] in the y-direction. Each bin is inclusive at the lower
end and exclusive at the upper end. Mathematically this means that the
bins are defined by the following inequality,
bin(i,j) corresponds to xrange[i] <= x < xrange[i+1]
and yrange[j] <= y < yrange[j+1]
Note that any samples which fall on the upper sides of the histogram are excluded. If you want to include these values for the side bins you will need to add an extra row or column to your histogram.
The gsl_histogram2d struct and its associated functions are
defined in the header file gsl_histogram2d.h.
Next: Using the library, Previous: Top, Up: Top [Index]
The GNU Scientific Library (GSL) is a collection of routines for numerical computing. The routines have been written from scratch in C, and present a modern Applications Programming Interface (API) for C programmers, allowing wrappers to be written for very high level languages. The source code is distributed under the GNU General Public License.
| • Routines available in GSL: | ||
| • GSL is Free Software: | ||
| • Obtaining GSL: | ||
| • No Warranty: | ||
| • Reporting Bugs: | ||
| • Further Information: | ||
| • Conventions used in this manual: |
Next: Linear Algebra References and Further Reading, Previous: Balancing, Up: Linear Algebra [Index]
The following program solves the linear system A x = b. The system to be solved is,
[ 0.18 0.60 0.57 0.96 ] [x0] [1.0] [ 0.41 0.24 0.99 0.58 ] [x1] = [2.0] [ 0.14 0.30 0.97 0.66 ] [x2] [3.0] [ 0.51 0.13 0.19 0.85 ] [x3] [4.0]
and the solution is found using LU decomposition of the matrix A.
#include <stdio.h>
#include <gsl/gsl_linalg.h>
int
main (void)
{
double a_data[] = { 0.18, 0.60, 0.57, 0.96,
0.41, 0.24, 0.99, 0.58,
0.14, 0.30, 0.97, 0.66,
0.51, 0.13, 0.19, 0.85 };
double b_data[] = { 1.0, 2.0, 3.0, 4.0 };
gsl_matrix_view m
= gsl_matrix_view_array (a_data, 4, 4);
gsl_vector_view b
= gsl_vector_view_array (b_data, 4);
gsl_vector *x = gsl_vector_alloc (4);
int s;
gsl_permutation * p = gsl_permutation_alloc (4);
gsl_linalg_LU_decomp (&m.matrix, p, &s);
gsl_linalg_LU_solve (&m.matrix, p, &b.vector, x);
printf ("x = \n");
gsl_vector_fprintf (stdout, x, "%g");
gsl_permutation_free (p);
gsl_vector_free (x);
return 0;
}
Here is the output from the program,
x = -4.05205 -12.6056 1.66091 8.69377
This can be verified by multiplying the solution x by the original matrix A using GNU OCTAVE,
octave> A = [ 0.18, 0.60, 0.57, 0.96;
0.41, 0.24, 0.99, 0.58;
0.14, 0.30, 0.97, 0.66;
0.51, 0.13, 0.19, 0.85 ];
octave> x = [ -4.05205; -12.6056; 1.66091; 8.69377];
octave> A * x
ans =
1.0000
2.0000
3.0000
4.0000
This reproduces the original right-hand side vector, b, in accordance with the equation A x = b.
gsl-ref-html-2.3/Sparse-Matrices-Exchanging-Rows-and-Columns.html 0000664 0001750 0001750 00000012406 13055414541 023007 0 ustar edd eddNext: Sparse Matrices Operations, Previous: Sparse Matrices Copying, Up: Sparse Matrices [Index]
This function copies the transpose of the sparse matrix src into dest. The dimensions of dest must match the transpose of the matrix src. Also, both matrices must use the same sparse storage format.
This function replaces the matrix m by its transpose, preserving the storage format of the input matrix. Currently, only triplet matrix inputs are supported.
This function replaces the matrix m by its transpose, but changes the storage format for compressed matrix inputs. Since compressed column storage is the transpose of compressed row storage, this function simply converts a CCS matrix to CRS and vice versa. This is the most efficient way to transpose a compressed storage matrix, but the user should note that the storage format of their compressed matrix will change on output. For triplet matrices, the output matrix is also in triplet storage.
Next: Discrete Hankel Transform References, Previous: Discrete Hankel Transform Definition, Up: Discrete Hankel Transforms [Index]
This function allocates a Discrete Hankel transform object of size size.
This function initializes the transform t for the given values of nu and xmax.
This function allocates a Discrete Hankel transform object of size size and initializes it for the given values of nu and xmax.
This function applies the transform t to the array f_in whose size is equal to the size of the transform. The result is stored in the array f_out which must be of the same length.
Applying this function to its output gives the original data multiplied by (1/j_(\nu,M))^2, up to numerical errors.
This function returns the value of the n-th sample point in the unit interval, (j_{\nu,n+1}/j_{\nu,M}) X. These are the points where the function f(t) is assumed to be sampled.
This function returns the value of the n-th sample point in “k-space”, j_{\nu,n+1}/X.
Next: Aliasing of arrays, Previous: Support for different numeric types, Up: Using the library [Index]
The library header files automatically define functions to have
extern "C" linkage when included in C++ programs. This allows
the functions to be called directly from C++.
To use C++ exception handling within user-defined functions passed to
the library as parameters, the library must be built with the
additional CFLAGS compilation option -fexceptions.
The functions described in this section can be used to perform least-squares fits to a straight line model, Y(c,x) = c_0 + c_1 x.
This function computes the best-fit linear regression coefficients
(c0,c1) of the model Y = c_0 + c_1 X for the dataset
(x, y), two vectors of length n with strides
xstride and ystride. The errors on y are assumed unknown so
the variance-covariance matrix for the
parameters (c0, c1) is estimated from the scatter of the
points around the best-fit line and returned via the parameters
(cov00, cov01, cov11).
The sum of squares of the residuals from the best-fit line is returned
in sumsq. Note: the correlation coefficient of the data can be computed using gsl_stats_correlation (see Correlation), it does not depend on the fit.
This function computes the best-fit linear regression coefficients (c0,c1) of the model Y = c_0 + c_1 X for the weighted dataset (x, y), two vectors of length n with strides xstride and ystride. The vector w, of length n and stride wstride, specifies the weight of each datapoint. The weight is the reciprocal of the variance for each datapoint in y.
The covariance matrix for the parameters (c0, c1) is computed using the weights and returned via the parameters (cov00, cov01, cov11). The weighted sum of squares of the residuals from the best-fit line, \chi^2, is returned in chisq.
This function uses the best-fit linear regression coefficients c0, c1 and their covariance cov00, cov01, cov11 to compute the fitted function y and its standard deviation y_err for the model Y = c_0 + c_1 X at the point x.
Next: Level 2 GSL BLAS Interface, Up: GSL BLAS Interface [Index]
This function computes the sum \alpha + x^T y for the vectors x and y, returning the result in result.
These functions compute the scalar product x^T y for the vectors x and y, returning the result in result.
These functions compute the complex scalar product x^T y for the vectors x and y, returning the result in dotu
These functions compute the complex conjugate scalar product x^H y for the vectors x and y, returning the result in dotc
These functions compute the Euclidean norm ||x||_2 = \sqrt {\sum x_i^2} of the vector x.
These functions compute the Euclidean norm of the complex vector x,
||x||_2 = \sqrt {\sum (\Re(x_i)^2 + \Im(x_i)^2)}.
These functions compute the absolute sum \sum |x_i| of the elements of the vector x.
These functions compute the sum of the magnitudes of the real and imaginary parts of the complex vector x, \sum |\Re(x_i)| + |\Im(x_i)|.
These functions return the index of the largest element of the vector x. The largest element is determined by its absolute magnitude for real vectors and by the sum of the magnitudes of the real and imaginary parts |\Re(x_i)| + |\Im(x_i)| for complex vectors. If the largest value occurs several times then the index of the first occurrence is returned.
These functions exchange the elements of the vectors x and y.
These functions copy the elements of the vector x into the vector y.
These functions compute the sum y = \alpha x + y for the vectors x and y.
These functions rescale the vector x by the multiplicative factor alpha.
These functions compute a Givens rotation (c,s) which zeroes the vector (a,b),
[ c s ] [ a ] = [ r ] [ -s c ] [ b ] [ 0 ]
The variables a and b are overwritten by the routine.
These functions apply a Givens rotation (x', y') = (c x + s y, -s x + c y) to the vectors x, y.
These functions compute a modified Givens transformation. The modified Givens transformation is defined in the original Level-1 BLAS specification, given in the references.
These functions apply a modified Givens transformation.
Next: Level 2 GSL BLAS Interface, Up: GSL BLAS Interface [Index]
Next: Inline functions, Previous: Shared Libraries, Up: Using the library [Index]
The library is written in ANSI C and is intended to conform to the ANSI C standard (C89). It should be portable to any system with a working ANSI C compiler.
The library does not rely on any non-ANSI extensions in the interface it exports to the user. Programs you write using GSL can be ANSI compliant. Extensions which can be used in a way compatible with pure ANSI C are supported, however, via conditional compilation. This allows the library to take advantage of compiler extensions on those platforms which support them.
When an ANSI C feature is known to be broken on a particular system the library will exclude any related functions at compile-time. This should make it impossible to link a program that would use these functions and give incorrect results.
To avoid namespace conflicts all exported function names and variables
have the prefix gsl_, while exported macros have the prefix
GSL_.
Next: Regular Modified Spherical Bessel Functions, Previous: Regular Spherical Bessel Functions, Up: Bessel Functions [Index]
These routines compute the irregular spherical Bessel function of zeroth order, y_0(x) = -\cos(x)/x.
These routines compute the irregular spherical Bessel function of first order, y_1(x) = -(\cos(x)/x + \sin(x))/x.
These routines compute the irregular spherical Bessel function of second order, y_2(x) = (-3/x^3 + 1/x)\cos(x) - (3/x^2)\sin(x).
These routines compute the irregular spherical Bessel function of order l, y_l(x), for l >= 0.
This routine computes the values of the irregular spherical Bessel functions y_l(x) for l from 0 to lmax inclusive for lmax >= 0, storing the results in the array result_array. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
Next: ODE Example programs, Previous: Evolution, Up: Ordinary Differential Equations [Index]
The driver object is a high level wrapper that combines the evolution, control and stepper objects for easy use.
These functions return a pointer to a newly allocated instance of a
driver object. The functions automatically allocate and initialise the
evolve, control and stepper objects for ODE system sys using
stepper type T. The initial step size is given in
hstart. The rest of the arguments follow the syntax and
semantics of the control functions with same name
(gsl_odeiv2_control_*_new).
The function sets a minimum for allowed step size hmin for driver d. Default value is 0.
The function sets a maximum for allowed step size hmax for
driver d. Default value is GSL_DBL_MAX.
The function sets a maximum for allowed number of steps nmax for driver d. Default value of 0 sets no limit for steps.
This function evolves the driver system d from t to
t1. Initially vector y should contain the values of
dependent variables at point t. If the function is unable to
complete the calculation, an error code from
gsl_odeiv2_evolve_apply is returned, and t and y
contain the values from last successful step.
If maximum number of steps is reached, a value of GSL_EMAXITER
is returned. If the step size drops below minimum value, the function
returns with GSL_ENOPROG. If the user-supplied functions
defined in the system sys returns GSL_EBADFUNC, the
function returns immediately with the same return code. In this case
the user must call gsl_odeiv2_driver_reset before calling this
function again.
This function evolves the driver system d from t with
n steps of size h. If the function is unable to complete
the calculation, an error code from
gsl_odeiv2_evolve_apply_fixed_step is returned, and t and
y contain the values from last successful step.
This function resets the evolution and stepper objects.
The routine resets the evolution and stepper objects and sets new initial step size to hstart. This function can be used e.g. to change the direction of integration.
This function frees the driver object, and the related evolution, stepper and control objects.
Next: ODE Example programs, Previous: Evolution, Up: Ordinary Differential Equations [Index]
Previous: Setting up your IEEE environment, Up: IEEE floating-point arithmetic [Index]
The reference for the IEEE standard is,
A more pedagogical introduction to the standard can be found in the following paper,
Corrigendum: ACM Computing Surveys, Vol. 23, No. 3 (September 1991), page 413. and see also the sections by B. A. Wichmann and Charles B. Dunham in Surveyor’s Forum: “What Every Computer Scientist Should Know About Floating-Point Arithmetic”. ACM Computing Surveys, Vol. 24, No. 3 (September 1992), page 319.
A detailed textbook on IEEE arithmetic and its practical use is available from SIAM Press,
Next: Nonlinear Least-Squares TRS Levenberg-Marquardt with Geodesic Acceleration, Up: Nonlinear Least-Squares TRS Overview [Index]
There is a theorem which states that if \delta_k is a solution to the trust region subproblem given above, then there exists \mu_k \ge 0 such that
( B_k + \mu_k D_k^T D_k ) \delta_k = -g_k
with \mu_k (\Delta_k - ||D_k \delta_k||) = 0. This forms the basis of the Levenberg-Marquardt algorithm, which controls the trust region size by adjusting the parameter \mu_k rather than the radius \Delta_k directly. For each radius \Delta_k, there is a unique parameter \mu_k which solves the TRS, and they have an inverse relationship, so that large values of \mu_k correspond to smaller trust regions, while small values of \mu_k correspond to larger trust regions.
With the approximation B_k \approx J_k^T J_k, on each iteration, in order to calculate the step \delta_k, the following linear least squares problem is solved:
[J_k; sqrt(mu_k) D_k] \delta_k = - [f_k; 0]
If the step \delta_k is accepted, then \mu_k is decreased on the next iteration in order to take a larger step, otherwise it is increased to take a smaller step. The Levenberg-Marquardt algorithm provides an exact solution of the trust region subproblem, but typically has a higher computational cost per iteration than the approximate methods discussed below, since it may need to solve the least squares system above several times for different values of \mu_k.
gsl-ref-html-2.3/Initializing-the-Minimizer.html 0000664 0001750 0001750 00000015271 13055414471 020004 0 ustar edd eddNext: Providing the function to minimize, Previous: Minimization Caveats, Up: One dimensional Minimization [Index]
This function returns a pointer to a newly allocated instance of a minimizer of type T. For example, the following code creates an instance of a golden section minimizer,
const gsl_min_fminimizer_type * T = gsl_min_fminimizer_goldensection; gsl_min_fminimizer * s = gsl_min_fminimizer_alloc (T);
If there is insufficient memory to create the minimizer then the function
returns a null pointer and the error handler is invoked with an error
code of GSL_ENOMEM.
This function sets, or resets, an existing minimizer s to use the function f and the initial search interval [x_lower, x_upper], with a guess for the location of the minimum x_minimum.
If the interval given does not contain a minimum, then the function
returns an error code of GSL_EINVAL.
This function is equivalent to gsl_min_fminimizer_set but uses
the values f_minimum, f_lower and f_upper instead of
computing f(x_minimum), f(x_lower) and f(x_upper).
This function frees all the memory associated with the minimizer s.
This function returns a pointer to the name of the minimizer. For example,
printf ("s is a '%s' minimizer\n",
gsl_min_fminimizer_name (s));
would print something like s is a 'brent' minimizer.
Next: Pivoted Cholesky Decomposition, Previous: Singular Value Decomposition, Up: Linear Algebra [Index]
A symmetric, positive definite square matrix A has a Cholesky decomposition into a product of a lower triangular matrix L and its transpose L^T,
A = L L^T
This is sometimes referred to as taking the square-root of a matrix. The Cholesky decomposition can only be carried out when all the eigenvalues of the matrix are positive. This decomposition can be used to convert the linear system A x = b into a pair of triangular systems (L y = b, L^T x = y), which can be solved by forward and back-substitution.
If the matrix A is near singular, it is sometimes possible to reduce the condition number and recover a more accurate solution vector x by scaling as
( S A S ) ( S^(-1) x ) = S b
where S is a diagonal matrix whose elements are given by S_{ii} = 1/\sqrt{A_{ii}}. This scaling is also known as Jacobi preconditioning. There are routines below to solve both the scaled and unscaled systems.
These functions factorize the symmetric, positive-definite square matrix
A into the Cholesky decomposition A = L L^T (or
A = L L^H
for the complex case). On input, the values from the diagonal and lower-triangular
part of the matrix A are used (the upper triangular part is ignored). On output
the diagonal and lower triangular part of the input matrix A contain the matrix
L, while the upper triangular part is unmodified. If the matrix is not
positive-definite then the decomposition will fail, returning the
error code GSL_EDOM.
When testing whether a matrix is positive-definite, disable the error handler first to avoid triggering an error.
This function is now deprecated and is provided only for backward compatibility.
These functions solve the system A x = b using the Cholesky
decomposition of A held in the matrix cholesky which must
have been previously computed by gsl_linalg_cholesky_decomp or
gsl_linalg_complex_cholesky_decomp.
These functions solve the system A x = b in-place using the
Cholesky decomposition of A held in the matrix cholesky
which must have been previously computed by
gsl_linalg_cholesky_decomp or
gsl_linalg_complex_cholesky_decomp. On input x should
contain the right-hand side b, which is replaced by the
solution on output.
These functions compute the inverse of a matrix from its Cholesky
decomposition cholesky, which must have been previously computed
by gsl_linalg_cholesky_decomp or
gsl_linalg_complex_cholesky_decomp. On output, the inverse is
stored in-place in cholesky.
This function calculates a diagonal scaling transformation S for
the symmetric, positive-definite square matrix A, and then
computes the Cholesky decomposition S A S = L L^T.
On input, the values from the diagonal and lower-triangular part of the matrix A are
used (the upper triangular part is ignored). On output the diagonal and lower triangular part
of the input matrix A contain the matrix L, while the upper triangular part
of the input matrix is overwritten with L^T (the diagonal terms being
identical for both L and L^T). If the matrix is not
positive-definite then the decomposition will fail, returning the
error code GSL_EDOM. The diagonal scale factors are stored in S
on output.
When testing whether a matrix is positive-definite, disable the error handler first to avoid triggering an error.
This function solves the system (S A S) (S^{-1} x) = S b using the Cholesky
decomposition of S A S held in the matrix cholesky which must
have been previously computed by gsl_linalg_cholesky_decomp2.
This function solves the system (S A S) (S^{-1} x) = S b in-place using the
Cholesky decomposition of S A S held in the matrix cholesky
which must have been previously computed by
gsl_linalg_cholesky_decomp2. On input x should
contain the right-hand side b, which is replaced by the
solution on output.
This function calculates a diagonal scaling transformation of the symmetric, positive definite matrix A, such that S A S has a condition number within a factor of N of the matrix of smallest possible condition number over all possible diagonal scalings. On output, S contains the scale factors, given by S_i = 1/\sqrt{A_{ii}}. For any A_{ii} \le 0, the corresponding scale factor S_i is set to 1.
This function applies the scaling transformation S to the matrix A. On output, A is replaced by S A S.
This function estimates the reciprocal condition number (using the 1-norm) of the symmetric positive definite matrix A, using its Cholesky decomposition provided in cholesky. The reciprocal condition number estimate, defined as 1 / (||A||_1 \cdot ||A^{-1}||_1), is stored in rcond. Additional workspace of size 3 N is required in work.
Next: Pivoted Cholesky Decomposition, Previous: Singular Value Decomposition, Up: Linear Algebra [Index]
These routines compute the Airy function Ai(x) with an accuracy specified by mode.
These routines compute the Airy function Bi(x) with an accuracy specified by mode.
These routines compute a scaled version of the Airy function S_A(x) Ai(x). For x>0 the scaling factor S_A(x) is \exp(+(2/3) x^(3/2)), and is 1 for x<0.
These routines compute a scaled version of the Airy function S_B(x) Bi(x). For x>0 the scaling factor S_B(x) is exp(-(2/3) x^(3/2)), and is 1 for x<0.
Next: Minimization Algorithms, Previous: Minimization Iteration, Up: One dimensional Minimization [Index]
A minimization procedure should stop when one of the following conditions is true:
The handling of these conditions is under user control. The function below allows the user to test the precision of the current result.
This function tests for the convergence of the interval [x_lower,
x_upper] with absolute error epsabs and relative error
epsrel. The test returns GSL_SUCCESS if the following
condition is achieved,
|a - b| < epsabs + epsrel min(|a|,|b|)
when the interval x = [a,b] does not include the origin. If the interval includes the origin then \min(|a|,|b|) is replaced by zero (which is the minimum value of |x| over the interval). This ensures that the relative error is accurately estimated for minima close to the origin.
This condition on the interval also implies that any estimate of the minimum x_m in the interval satisfies the same condition with respect to the true minimum x_m^*,
|x_m - x_m^*| < epsabs + epsrel x_m^*
assuming that the true minimum x_m^* is contained within the interval.
Next: Inverse Complex Hyperbolic Functions, Previous: Inverse Complex Trigonometric Functions, Up: Complex Numbers [Index]
This function returns the complex hyperbolic sine of the complex number z, \sinh(z) = (\exp(z) - \exp(-z))/2.
This function returns the complex hyperbolic cosine of the complex number z, \cosh(z) = (\exp(z) + \exp(-z))/2.
This function returns the complex hyperbolic tangent of the complex number z, \tanh(z) = \sinh(z)/\cosh(z).
This function returns the complex hyperbolic secant of the complex number z, \sech(z) = 1/\cosh(z).
This function returns the complex hyperbolic cosecant of the complex number z, \csch(z) = 1/\sinh(z).
This function returns the complex hyperbolic cotangent of the complex number z, \coth(z) = 1/\tanh(z).
Previous: Matrix properties, Up: Matrices [Index]
The program below shows how to allocate, initialize and read from a matrix
using the functions gsl_matrix_alloc, gsl_matrix_set and
gsl_matrix_get.
#include <stdio.h>
#include <gsl/gsl_matrix.h>
int
main (void)
{
int i, j;
gsl_matrix * m = gsl_matrix_alloc (10, 3);
for (i = 0; i < 10; i++)
for (j = 0; j < 3; j++)
gsl_matrix_set (m, i, j, 0.23 + 100*i + j);
for (i = 0; i < 100; i++) /* OUT OF RANGE ERROR */
for (j = 0; j < 3; j++)
printf ("m(%d,%d) = %g\n", i, j,
gsl_matrix_get (m, i, j));
gsl_matrix_free (m);
return 0;
}
Here is the output from the program. The final loop attempts to read
outside the range of the matrix m, and the error is trapped by
the range-checking code in gsl_matrix_get.
$ ./a.out m(0,0) = 0.23 m(0,1) = 1.23 m(0,2) = 2.23 m(1,0) = 100.23 m(1,1) = 101.23 m(1,2) = 102.23 ... m(9,2) = 902.23 gsl: matrix_source.c:13: ERROR: first index out of range Default GSL error handler invoked. Aborted (core dumped)
The next program shows how to write a matrix to a file.
#include <stdio.h>
#include <gsl/gsl_matrix.h>
int
main (void)
{
int i, j, k = 0;
gsl_matrix * m = gsl_matrix_alloc (100, 100);
gsl_matrix * a = gsl_matrix_alloc (100, 100);
for (i = 0; i < 100; i++)
for (j = 0; j < 100; j++)
gsl_matrix_set (m, i, j, 0.23 + i + j);
{
FILE * f = fopen ("test.dat", "wb");
gsl_matrix_fwrite (f, m);
fclose (f);
}
{
FILE * f = fopen ("test.dat", "rb");
gsl_matrix_fread (f, a);
fclose (f);
}
for (i = 0; i < 100; i++)
for (j = 0; j < 100; j++)
{
double mij = gsl_matrix_get (m, i, j);
double aij = gsl_matrix_get (a, i, j);
if (mij != aij) k++;
}
gsl_matrix_free (m);
gsl_matrix_free (a);
printf ("differences = %d (should be zero)\n", k);
return (k > 0);
}
After running this program the file test.dat should contain the
elements of m, written in binary format. The matrix which is read
back in using the function gsl_matrix_fread should be exactly
equal to the original matrix.
The following program demonstrates the use of vector views. The program computes the column norms of a matrix.
#include <math.h>
#include <stdio.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
int
main (void)
{
size_t i,j;
gsl_matrix *m = gsl_matrix_alloc (10, 10);
for (i = 0; i < 10; i++)
for (j = 0; j < 10; j++)
gsl_matrix_set (m, i, j, sin (i) + cos (j));
for (j = 0; j < 10; j++)
{
gsl_vector_view column = gsl_matrix_column (m, j);
double d;
d = gsl_blas_dnrm2 (&column.vector);
printf ("matrix column %zu, norm = %g\n", j, d);
}
gsl_matrix_free (m);
return 0;
}
Here is the output of the program,
$ ./a.out
matrix column 0, norm = 4.31461 matrix column 1, norm = 3.1205 matrix column 2, norm = 2.19316 matrix column 3, norm = 3.26114 matrix column 4, norm = 2.53416 matrix column 5, norm = 2.57281 matrix column 6, norm = 4.20469 matrix column 7, norm = 3.65202 matrix column 8, norm = 2.08524 matrix column 9, norm = 3.07313
The results can be confirmed using GNU OCTAVE,
$ octave
GNU Octave, version 2.0.16.92
octave> m = sin(0:9)' * ones(1,10)
+ ones(10,1) * cos(0:9);
octave> sqrt(sum(m.^2))
ans =
4.3146 3.1205 2.1932 3.2611 2.5342 2.5728
4.2047 3.6520 2.0852 3.0731
Previous: Matrix properties, Up: Matrices [Index]
Next: Fitting large linear systems example, Previous: Fitting regularized linear regression example 2, Up: Fitting Examples [Index]
The next program demonstrates the advantage of robust least squares on a dataset with outliers. The program generates linear (x,y) data pairs on the line y = 1.45 x + 3.88, adds some random noise, and inserts 3 outliers into the dataset. Both the robust and ordinary least squares (OLS) coefficients are computed for comparison.
#include <stdio.h>
#include <gsl/gsl_multifit.h>
#include <gsl/gsl_randist.h>
int
dofit(const gsl_multifit_robust_type *T,
const gsl_matrix *X, const gsl_vector *y,
gsl_vector *c, gsl_matrix *cov)
{
int s;
gsl_multifit_robust_workspace * work
= gsl_multifit_robust_alloc (T, X->size1, X->size2);
s = gsl_multifit_robust (X, y, c, cov, work);
gsl_multifit_robust_free (work);
return s;
}
int
main (int argc, char **argv)
{
size_t i;
size_t n;
const size_t p = 2; /* linear fit */
gsl_matrix *X, *cov;
gsl_vector *x, *y, *c, *c_ols;
const double a = 1.45; /* slope */
const double b = 3.88; /* intercept */
gsl_rng *r;
if (argc != 2)
{
fprintf (stderr,"usage: robfit n\n");
exit (-1);
}
n = atoi (argv[1]);
X = gsl_matrix_alloc (n, p);
x = gsl_vector_alloc (n);
y = gsl_vector_alloc (n);
c = gsl_vector_alloc (p);
c_ols = gsl_vector_alloc (p);
cov = gsl_matrix_alloc (p, p);
r = gsl_rng_alloc(gsl_rng_default);
/* generate linear dataset */
for (i = 0; i < n - 3; i++)
{
double dx = 10.0 / (n - 1.0);
double ei = gsl_rng_uniform(r);
double xi = -5.0 + i * dx;
double yi = a * xi + b;
gsl_vector_set (x, i, xi);
gsl_vector_set (y, i, yi + ei);
}
/* add a few outliers */
gsl_vector_set(x, n - 3, 4.7);
gsl_vector_set(y, n - 3, -8.3);
gsl_vector_set(x, n - 2, 3.5);
gsl_vector_set(y, n - 2, -6.7);
gsl_vector_set(x, n - 1, 4.1);
gsl_vector_set(y, n - 1, -6.0);
/* construct design matrix X for linear fit */
for (i = 0; i < n; ++i)
{
double xi = gsl_vector_get(x, i);
gsl_matrix_set (X, i, 0, 1.0);
gsl_matrix_set (X, i, 1, xi);
}
/* perform robust and OLS fit */
dofit(gsl_multifit_robust_ols, X, y, c_ols, cov);
dofit(gsl_multifit_robust_bisquare, X, y, c, cov);
/* output data and model */
for (i = 0; i < n; ++i)
{
double xi = gsl_vector_get(x, i);
double yi = gsl_vector_get(y, i);
gsl_vector_view v = gsl_matrix_row(X, i);
double y_ols, y_rob, y_err;
gsl_multifit_robust_est(&v.vector, c, cov, &y_rob, &y_err);
gsl_multifit_robust_est(&v.vector, c_ols, cov, &y_ols, &y_err);
printf("%g %g %g %g\n", xi, yi, y_rob, y_ols);
}
#define C(i) (gsl_vector_get(c,(i)))
#define COV(i,j) (gsl_matrix_get(cov,(i),(j)))
{
printf ("# best fit: Y = %g + %g X\n",
C(0), C(1));
printf ("# covariance matrix:\n");
printf ("# [ %+.5e, %+.5e\n",
COV(0,0), COV(0,1));
printf ("# %+.5e, %+.5e\n",
COV(1,0), COV(1,1));
}
gsl_matrix_free (X);
gsl_vector_free (x);
gsl_vector_free (y);
gsl_vector_free (c);
gsl_vector_free (c_ols);
gsl_matrix_free (cov);
gsl_rng_free(r);
return 0;
}
The output from the program is shown in the following plot.
Next: Fitting large linear systems example, Previous: Fitting regularized linear regression example 2, Up: Fitting Examples [Index]
Next: Associated Legendre Polynomials and Spherical Harmonics, Up: Legendre Functions and Spherical Harmonics [Index]
These functions evaluate the Legendre polynomials P_l(x) using explicit representations for l=1, 2, 3.
These functions evaluate the Legendre polynomial P_l(x) for a specific value of l, x subject to l >= 0, |x| <= 1
These functions compute arrays of Legendre polynomials P_l(x) and derivatives dP_l(x)/dx, for l = 0, \dots, lmax, |x| <= 1
These routines compute the Legendre function Q_0(x) for x > -1, x != 1.
These routines compute the Legendre function Q_1(x) for x > -1, x != 1.
These routines compute the Legendre function Q_l(x) for x > -1, x != 1 and l >= 0.
Next: Fitting regularized linear regression example 2, Previous: Fitting multi-parameter linear regression example, Up: Fitting Examples [Index]
The next program demonstrates the difference between ordinary and regularized least squares when the design matrix is near-singular. In this program, we generate two random normally distributed variables u and v, with v = u + noise so that u and v are nearly colinear. We then set a third dependent variable y = u + v + noise and solve for the coefficients c_1,c_2 of the model Y(c_1,c_2) = c_1 u + c_2 v. Since u \approx v, the design matrix X is nearly singular, leading to unstable ordinary least squares solutions.
Here is the program output:
matrix condition number = 1.025113e+04 === Unregularized fit === best fit: y = -43.6588 u + 45.6636 v residual norm = 31.6248 solution norm = 63.1764 chisq/dof = 1.00213 === Regularized fit (L-curve) === optimal lambda: 4.51103 best fit: y = 1.00113 u + 1.0032 v residual norm = 31.6547 solution norm = 1.41728 chisq/dof = 1.04499 === Regularized fit (GCV) === optimal lambda: 0.0232029 best fit: y = -19.8367 u + 21.8417 v residual norm = 31.6332 solution norm = 29.5051 chisq/dof = 1.00314
We see that the ordinary least squares solution is completely wrong, while the L-curve regularized method with the optimal \lambda = 4.51103 finds the correct solution c_1 \approx c_2 \approx 1. The GCV regularized method finds a regularization parameter \lambda = 0.0232029 which is too small to give an accurate solution, although it performs better than OLS. The L-curve and its computed corner, as well as the GCV curve and its minimum are plotted below.
The program is given below.
#include <gsl/gsl_math.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_multifit.h>
int
main()
{
const size_t n = 1000; /* number of observations */
const size_t p = 2; /* number of model parameters */
size_t i;
gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
gsl_matrix *X = gsl_matrix_alloc(n, p);
gsl_vector *y = gsl_vector_alloc(n);
for (i = 0; i < n; ++i)
{
/* generate first random variable u */
double ui = 5.0 * gsl_ran_gaussian(r, 1.0);
/* set v = u + noise */
double vi = ui + gsl_ran_gaussian(r, 0.001);
/* set y = u + v + noise */
double yi = ui + vi + gsl_ran_gaussian(r, 1.0);
/* since u =~ v, the matrix X is ill-conditioned */
gsl_matrix_set(X, i, 0, ui);
gsl_matrix_set(X, i, 1, vi);
/* rhs vector */
gsl_vector_set(y, i, yi);
}
{
const size_t npoints = 200; /* number of points on L-curve and GCV curve */
gsl_multifit_linear_workspace *w =
gsl_multifit_linear_alloc(n, p);
gsl_vector *c = gsl_vector_alloc(p); /* OLS solution */
gsl_vector *c_lcurve = gsl_vector_alloc(p); /* regularized solution (L-curve) */
gsl_vector *c_gcv = gsl_vector_alloc(p); /* regularized solution (GCV) */
gsl_vector *reg_param = gsl_vector_alloc(npoints);
gsl_vector *rho = gsl_vector_alloc(npoints); /* residual norms */
gsl_vector *eta = gsl_vector_alloc(npoints); /* solution norms */
gsl_vector *G = gsl_vector_alloc(npoints); /* GCV function values */
double lambda_l; /* optimal regularization parameter (L-curve) */
double lambda_gcv; /* optimal regularization parameter (GCV) */
double G_gcv; /* G(lambda_gcv) */
size_t reg_idx; /* index of optimal lambda */
double rcond; /* reciprocal condition number of X */
double chisq, rnorm, snorm;
/* compute SVD of X */
gsl_multifit_linear_svd(X, w);
rcond = gsl_multifit_linear_rcond(w);
fprintf(stderr, "matrix condition number = %e\n", 1.0 / rcond);
/* unregularized (standard) least squares fit, lambda = 0 */
gsl_multifit_linear_solve(0.0, X, y, c, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0);
fprintf(stderr, "=== Unregularized fit ===\n");
fprintf(stderr, "best fit: y = %g u + %g v\n",
gsl_vector_get(c, 0), gsl_vector_get(c, 1));
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));
/* calculate L-curve and find its corner */
gsl_multifit_linear_lcurve(y, reg_param, rho, eta, w);
gsl_multifit_linear_lcorner(rho, eta, ®_idx);
/* store optimal regularization parameter */
lambda_l = gsl_vector_get(reg_param, reg_idx);
/* regularize with lambda_l */
gsl_multifit_linear_solve(lambda_l, X, y, c_lcurve, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);
fprintf(stderr, "=== Regularized fit (L-curve) ===\n");
fprintf(stderr, "optimal lambda: %g\n", lambda_l);
fprintf(stderr, "best fit: y = %g u + %g v\n",
gsl_vector_get(c_lcurve, 0), gsl_vector_get(c_lcurve, 1));
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));
/* calculate GCV curve and find its minimum */
gsl_multifit_linear_gcv(y, reg_param, G, &lambda_gcv, &G_gcv, w);
/* regularize with lambda_gcv */
gsl_multifit_linear_solve(lambda_gcv, X, y, c_gcv, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0) + pow(lambda_gcv * snorm, 2.0);
fprintf(stderr, "=== Regularized fit (GCV) ===\n");
fprintf(stderr, "optimal lambda: %g\n", lambda_gcv);
fprintf(stderr, "best fit: y = %g u + %g v\n",
gsl_vector_get(c_gcv, 0), gsl_vector_get(c_gcv, 1));
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));
/* output L-curve and GCV curve */
for (i = 0; i < npoints; ++i)
{
printf("%e %e %e %e\n",
gsl_vector_get(reg_param, i),
gsl_vector_get(rho, i),
gsl_vector_get(eta, i),
gsl_vector_get(G, i));
}
/* output L-curve corner point */
printf("\n\n%f %f\n",
gsl_vector_get(rho, reg_idx),
gsl_vector_get(eta, reg_idx));
/* output GCV curve corner minimum */
printf("\n\n%e %e\n",
lambda_gcv,
G_gcv);
gsl_multifit_linear_free(w);
gsl_vector_free(c);
gsl_vector_free(c_lcurve);
gsl_vector_free(reg_param);
gsl_vector_free(rho);
gsl_vector_free(eta);
gsl_vector_free(G);
}
gsl_rng_free(r);
gsl_matrix_free(X);
gsl_vector_free(y);
return 0;
}
Next: Fitting regularized linear regression example 2, Previous: Fitting multi-parameter linear regression example, Up: Fitting Examples [Index]
Next: Complex arithmetic operators, Previous: Representation of complex numbers, Up: Complex Numbers [Index]
This function returns the argument of the complex number z, \arg(z), where -\pi < \arg(z) <= \pi.
This function returns the magnitude of the complex number z, |z|.
This function returns the squared magnitude of the complex number z, |z|^2.
This function returns the natural logarithm of the magnitude of the
complex number z, \log|z|. It allows an accurate
evaluation of \log|z| when |z| is close to one. The direct
evaluation of log(gsl_complex_abs(z)) would lead to a loss of
precision in this case.
Next: Atomic and Nuclear Physics, Previous: Fundamental Constants, Up: Physical Constants [Index]
GSL_CONST_MKSA_ASTRONOMICAL_UNITThe length of 1 astronomical unit (mean earth-sun distance), au.
GSL_CONST_MKSA_GRAVITATIONAL_CONSTANTThe gravitational constant, G.
GSL_CONST_MKSA_LIGHT_YEARThe distance of 1 light-year, ly.
GSL_CONST_MKSA_PARSECThe distance of 1 parsec, pc.
GSL_CONST_MKSA_GRAV_ACCELThe standard gravitational acceleration on Earth, g.
GSL_CONST_MKSA_SOLAR_MASSThe mass of the Sun.
Next: Householder Transformations, Previous: Bidiagonalization, Up: Linear Algebra [Index]
A Givens rotation is a rotation in the plane acting on two elements of a given vector. It can be represented in matrix form as
where the \cos{\theta} and \sin{\theta} appear at the intersection of the ith and jth rows and columns. When acting on a vector x, G(i,j,\theta) x performs a rotation of the (i,j) elements of x. Givens rotations are typically used to introduce zeros in vectors, such as during the QR decomposition of a matrix. In this case, it is typically desired to find c and s such that
with r = \sqrt{a^2 + b^2}.
This function computes c = \cos{\theta} and s = \sin{\theta} so that the Givens matrix G(\theta) acting on the vector (a,b) produces (r, 0), with r = \sqrt{a^2 + b^2}.
This function applies the Givens rotation defined by c = \cos{\theta} and s = \sin{\theta} to the i and j elements of v. On output, (v(i),v(j)) \leftarrow G(\theta) (v(i),v(j)).
Next: B-Spline References and Further Reading, Previous: Working with the Greville abscissae, Up: Basis Splines [Index]
The following program computes a linear least squares fit to data using cubic B-spline basis functions with uniform breakpoints. The data is generated from the curve y(x) = \cos{(x)} \exp{(-x/10)} on the interval [0, 15] with Gaussian noise added.
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <gsl/gsl_bspline.h>
#include <gsl/gsl_multifit.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_statistics.h>
/* number of data points to fit */
#define N 200
/* number of fit coefficients */
#define NCOEFFS 12
/* nbreak = ncoeffs + 2 - k = ncoeffs - 2 since k = 4 */
#define NBREAK (NCOEFFS - 2)
int
main (void)
{
const size_t n = N;
const size_t ncoeffs = NCOEFFS;
const size_t nbreak = NBREAK;
size_t i, j;
gsl_bspline_workspace *bw;
gsl_vector *B;
double dy;
gsl_rng *r;
gsl_vector *c, *w;
gsl_vector *x, *y;
gsl_matrix *X, *cov;
gsl_multifit_linear_workspace *mw;
double chisq, Rsq, dof, tss;
gsl_rng_env_setup();
r = gsl_rng_alloc(gsl_rng_default);
/* allocate a cubic bspline workspace (k = 4) */
bw = gsl_bspline_alloc(4, nbreak);
B = gsl_vector_alloc(ncoeffs);
x = gsl_vector_alloc(n);
y = gsl_vector_alloc(n);
X = gsl_matrix_alloc(n, ncoeffs);
c = gsl_vector_alloc(ncoeffs);
w = gsl_vector_alloc(n);
cov = gsl_matrix_alloc(ncoeffs, ncoeffs);
mw = gsl_multifit_linear_alloc(n, ncoeffs);
printf("#m=0,S=0\n");
/* this is the data to be fitted */
for (i = 0; i < n; ++i)
{
double sigma;
double xi = (15.0 / (N - 1)) * i;
double yi = cos(xi) * exp(-0.1 * xi);
sigma = 0.1 * yi;
dy = gsl_ran_gaussian(r, sigma);
yi += dy;
gsl_vector_set(x, i, xi);
gsl_vector_set(y, i, yi);
gsl_vector_set(w, i, 1.0 / (sigma * sigma));
printf("%f %f\n", xi, yi);
}
/* use uniform breakpoints on [0, 15] */
gsl_bspline_knots_uniform(0.0, 15.0, bw);
/* construct the fit matrix X */
for (i = 0; i < n; ++i)
{
double xi = gsl_vector_get(x, i);
/* compute B_j(xi) for all j */
gsl_bspline_eval(xi, B, bw);
/* fill in row i of X */
for (j = 0; j < ncoeffs; ++j)
{
double Bj = gsl_vector_get(B, j);
gsl_matrix_set(X, i, j, Bj);
}
}
/* do the fit */
gsl_multifit_wlinear(X, w, y, c, cov, &chisq, mw);
dof = n - ncoeffs;
tss = gsl_stats_wtss(w->data, 1, y->data, 1, y->size);
Rsq = 1.0 - chisq / tss;
fprintf(stderr, "chisq/dof = %e, Rsq = %f\n",
chisq / dof, Rsq);
/* output the smoothed curve */
{
double xi, yi, yerr;
printf("#m=1,S=0\n");
for (xi = 0.0; xi < 15.0; xi += 0.1)
{
gsl_bspline_eval(xi, B, bw);
gsl_multifit_linear_est(B, c, cov, &yi, &yerr);
printf("%f %f\n", xi, yi);
}
}
gsl_rng_free(r);
gsl_bspline_free(bw);
gsl_vector_free(B);
gsl_vector_free(x);
gsl_vector_free(y);
gsl_matrix_free(X);
gsl_vector_free(c);
gsl_vector_free(w);
gsl_matrix_free(cov);
gsl_multifit_linear_free(mw);
return 0;
} /* main() */
The output can be plotted with GNU graph.
$ ./a.out > bspline.txt chisq/dof = 1.118217e+00, Rsq = 0.989771 $ graph -T ps -X x -Y y -x 0 15 -y -1 1.3 < bspline.txt > bspline.ps
Next: B-Spline References and Further Reading, Previous: Working with the Greville abscissae, Up: Basis Splines [Index]
Next: Series Acceleration References, Previous: Acceleration functions without error estimation, Up: Series Acceleration [Index]
The following code calculates an estimate of \zeta(2) = \pi^2 / 6 using the series,
\zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + ...
After N terms the error in the sum is O(1/N), making direct summation of the series converge slowly.
#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_sum.h>
#define N 20
int
main (void)
{
double t[N];
double sum_accel, err;
double sum = 0;
int n;
gsl_sum_levin_u_workspace * w
= gsl_sum_levin_u_alloc (N);
const double zeta_2 = M_PI * M_PI / 6.0;
/* terms for zeta(2) = \sum_{n=1}^{\infty} 1/n^2 */
for (n = 0; n < N; n++)
{
double np1 = n + 1.0;
t[n] = 1.0 / (np1 * np1);
sum += t[n];
}
gsl_sum_levin_u_accel (t, N, w, &sum_accel, &err);
printf ("term-by-term sum = % .16f using %d terms\n",
sum, N);
printf ("term-by-term sum = % .16f using %zu terms\n",
w->sum_plain, w->terms_used);
printf ("exact value = % .16f\n", zeta_2);
printf ("accelerated sum = % .16f using %zu terms\n",
sum_accel, w->terms_used);
printf ("estimated error = % .16f\n", err);
printf ("actual error = % .16f\n",
sum_accel - zeta_2);
gsl_sum_levin_u_free (w);
return 0;
}
The output below shows that the Levin u-transform is able to obtain an estimate of the sum to 1 part in 10^10 using the first eleven terms of the series. The error estimate returned by the function is also accurate, giving the correct number of significant digits.
$ ./a.out
term-by-term sum = 1.5961632439130233 using 20 terms term-by-term sum = 1.5759958390005426 using 13 terms exact value = 1.6449340668482264 accelerated sum = 1.6449340669228176 using 13 terms estimated error = 0.0000000000888360 actual error = 0.0000000000745912
Note that a direct summation of this series would require 10^10 terms to achieve the same precision as the accelerated sum does in 13 terms.
Next: Series Acceleration References, Previous: Acceleration functions without error estimation, Up: Series Acceleration [Index]
Next: Roots of Polynomials Examples, Previous: Cubic Equations, Up: Polynomials [Index]
The roots of polynomial equations cannot be found analytically beyond the special cases of the quadratic, cubic and quartic equation. The algorithm described in this section uses an iterative method to find the approximate locations of roots of higher order polynomials.
This function allocates space for a gsl_poly_complex_workspace
struct and a workspace suitable for solving a polynomial with n
coefficients using the routine gsl_poly_complex_solve.
The function returns a pointer to the newly allocated
gsl_poly_complex_workspace if no errors were detected, and a null
pointer in the case of error.
This function frees all the memory associated with the workspace w.
This function computes the roots of the general polynomial P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1} using balanced-QR reduction of the companion matrix. The parameter n specifies the length of the coefficient array. The coefficient of the highest order term must be non-zero. The function requires a workspace w of the appropriate size. The n-1 roots are returned in the packed complex array z of length 2(n-1), alternating real and imaginary parts.
The function returns GSL_SUCCESS if all the roots are found. If
the QR reduction does not converge, the error handler is invoked with
an error code of GSL_EFAILED. Note that due to finite precision,
roots of higher multiplicity are returned as a cluster of simple roots
with reduced accuracy. The solution of polynomials with higher-order
roots requires specialized algorithms that take the multiplicity
structure into account (see e.g. Z. Zeng, Algorithm 835, ACM
Transactions on Mathematical Software, Volume 30, Issue 2 (2004), pp
218–236).
Next: Roots of Polynomials Examples, Previous: Cubic Equations, Up: Polynomials [Index]
Next: Complex Number References and Further Reading, Previous: Complex Hyperbolic Functions, Up: Complex Numbers [Index]
This function returns the complex hyperbolic arcsine of the complex number z, \arcsinh(z). The branch cuts are on the imaginary axis, below -i and above i.
This function returns the complex hyperbolic arccosine of the complex number z, \arccosh(z). The branch cut is on the real axis, less than 1. Note that in this case we use the negative square root in formula 4.6.21 of Abramowitz & Stegun giving \arccosh(z)=\log(z-\sqrt{z^2-1}).
This function returns the complex hyperbolic arccosine of the real number z, \arccosh(z).
This function returns the complex hyperbolic arctangent of the complex number z, \arctanh(z). The branch cuts are on the real axis, less than -1 and greater than 1.
This function returns the complex hyperbolic arctangent of the real number z, \arctanh(z).
This function returns the complex hyperbolic arcsecant of the complex number z, \arcsech(z) = \arccosh(1/z).
This function returns the complex hyperbolic arccosecant of the complex number z, \arccsch(z) = \arcsin(1/z).
This function returns the complex hyperbolic arccotangent of the complex number z, \arccoth(z) = \arctanh(1/z).
In 1988, Park and Miller wrote a paper entitled “Random number generators: good ones are hard to find.” [Commun. ACM, 31, 1192–1201]. Fortunately, some excellent random number generators are available, though poor ones are still in common use. You may be happy with the system-supplied random number generator on your computer, but you should be aware that as computers get faster, requirements on random number generators increase. Nowadays, a simulation that calls a random number generator millions of times can often finish before you can make it down the hall to the coffee machine and back.
A very nice review of random number generators was written by Pierre L’Ecuyer, as Chapter 4 of the book: Handbook on Simulation, Jerry Banks, ed. (Wiley, 1997). The chapter is available in postscript from L’Ecuyer’s ftp site (see references). Knuth’s volume on Seminumerical Algorithms (originally published in 1968) devotes 170 pages to random number generators, and has recently been updated in its 3rd edition (1997). It is brilliant, a classic. If you don’t own it, you should stop reading right now, run to the nearest bookstore, and buy it.
A good random number generator will satisfy both theoretical and statistical properties. Theoretical properties are often hard to obtain (they require real math!), but one prefers a random number generator with a long period, low serial correlation, and a tendency not to “fall mainly on the planes.” Statistical tests are performed with numerical simulations. Generally, a random number generator is used to estimate some quantity for which the theory of probability provides an exact answer. Comparison to this exact answer provides a measure of “randomness”.
Next: Regularized regression, Previous: Linear regression, Up: Least-Squares Fitting [Index]
This section describes routines which perform least squares fits to a linear model by minimizing the cost function
\chi^2 = \sum_i w_i (y_i - \sum_j X_ij c_j)^2 = || y - Xc ||_W^2
where y is a vector of n observations, X is an n-by-p matrix of predictor variables, c is a vector of the p unknown best-fit parameters to be estimated, and ||r||_W^2 = r^T W r. The matrix W = diag(w_1,w_2,...,w_n) defines the weights or uncertainties of the observation vector.
This formulation can be used for fits to any number of functions and/or variables by preparing the n-by-p matrix X appropriately. For example, to fit to a p-th order polynomial in x, use the following matrix,
X_{ij} = x_i^j
where the index i runs over the observations and the index j runs from 0 to p-1.
To fit to a set of p sinusoidal functions with fixed frequencies \omega_1, \omega_2, …, \omega_p, use,
X_{ij} = sin(\omega_j x_i)
To fit to p independent variables x_1, x_2, …, x_p, use,
X_{ij} = x_j(i)
where x_j(i) is the i-th value of the predictor variable x_j.
The solution of the general linear least-squares system requires an additional working space for intermediate results, such as the singular value decomposition of the matrix X.
These functions are declared in the header file gsl_multifit.h.
This function allocates a workspace for fitting a model to a maximum of n observations using a maximum of p parameters. The user may later supply a smaller least squares system if desired. The size of the workspace is O(np + p^2).
This function frees the memory associated with the workspace w.
This function performs a singular value decomposition of the matrix X and stores the SVD factors internally in work.
This function performs a singular value decomposition of the matrix X and stores the SVD factors internally in work. The matrix X is first balanced by applying column scaling factors to improve the accuracy of the singular values.
This function computes the best-fit parameters c of the model
y = X c for the observations y and the matrix of
predictor variables X, using the preallocated workspace provided
in work. The p-by-p variance-covariance matrix of the model parameters
cov is set to \sigma^2 (X^T X)^{-1}, where \sigma is
the standard deviation of the fit residuals.
The sum of squares of the residuals from the best-fit,
\chi^2, is returned in chisq. If the coefficient of
determination is desired, it can be computed from the expression
R^2 = 1 - \chi^2 / TSS, where the total sum of squares (TSS) of
the observations y may be computed from gsl_stats_tss.
The best-fit is found by singular value decomposition of the matrix X using the modified Golub-Reinsch SVD algorithm, with column scaling to improve the accuracy of the singular values. Any components which have zero singular value (to machine precision) are discarded from the fit.
This function computes the best-fit parameters c of the model
y = X c for the observations y and the matrix of
predictor variables X, using a truncated SVD expansion.
Singular values which satisfy s_i \le tol \times s_0
are discarded from the fit, where s_0 is the largest singular value.
The p-by-p variance-covariance matrix of the model parameters
cov is set to \sigma^2 (X^T X)^{-1}, where \sigma is
the standard deviation of the fit residuals.
The sum of squares of the residuals from the best-fit,
\chi^2, is returned in chisq. The effective rank
(number of singular values used in solution) is returned in rank.
If the coefficient of
determination is desired, it can be computed from the expression
R^2 = 1 - \chi^2 / TSS, where the total sum of squares (TSS) of
the observations y may be computed from gsl_stats_tss.
This function computes the best-fit parameters c of the weighted
model y = X c for the observations y with weights w
and the matrix of predictor variables X, using the preallocated
workspace provided in work. The p-by-p covariance matrix of the model
parameters cov is computed as (X^T W X)^{-1}. The weighted
sum of squares of the residuals from the best-fit, \chi^2, is
returned in chisq. If the coefficient of determination is
desired, it can be computed from the expression R^2 = 1 - \chi^2
/ WTSS, where the weighted total sum of squares (WTSS) of the
observations y may be computed from gsl_stats_wtss.
This function computes the best-fit parameters c of the weighted
model y = X c for the observations y with weights w
and the matrix of predictor variables X, using a truncated SVD expansion.
Singular values which satisfy s_i \le tol \times s_0
are discarded from the fit, where s_0 is the largest singular value.
The p-by-p covariance matrix of the model
parameters cov is computed as (X^T W X)^{-1}. The weighted
sum of squares of the residuals from the best-fit, \chi^2, is
returned in chisq. The effective rank of the system (number of
singular values used in the solution) is returned in rank.
If the coefficient of determination is
desired, it can be computed from the expression R^2 = 1 - \chi^2
/ WTSS, where the weighted total sum of squares (WTSS) of the
observations y may be computed from gsl_stats_wtss.
This function uses the best-fit multilinear regression coefficients c and their covariance matrix cov to compute the fitted function value y and its standard deviation y_err for the model y = x.c at the point x.
This function computes the vector of residuals r = y - X c for the observations y, coefficients c and matrix of predictor variables X.
This function returns the rank of the matrix X which must first have its singular value decomposition computed. The rank is computed by counting the number of singular values \sigma_j which satisfy \sigma_j > tol \times \sigma_0, where \sigma_0 is the largest singular value.
Next: Regularized regression, Previous: Linear regression, Up: Least-Squares Fitting [Index]
Next: Interpolation, Previous: Simulated Annealing, Up: Top [Index]
This chapter describes functions for solving ordinary differential equation (ODE) initial value problems. The library provides a variety of low-level methods, such as Runge-Kutta and Bulirsch-Stoer routines, and higher-level components for adaptive step-size control. The components can be combined by the user to achieve the desired solution, with full access to any intermediate steps. A driver object can be used as a high level wrapper for easy use of low level functions.
These functions are declared in the header file gsl_odeiv2.h.
This is a new interface in version 1.15 and uses the prefix
gsl_odeiv2 for all functions. It is recommended over the
previous gsl_odeiv implementation defined in gsl_odeiv.h
The old interface has been retained under the original name for
backwards compatibility.
| • Defining the ODE System: | ||
| • Stepping Functions: | ||
| • Adaptive Step-size Control: | ||
| • Evolution: | ||
| • Driver: | ||
| • ODE Example programs: | ||
| • ODE References and Further Reading: |
Next: Hyperbolic Integrals, Previous: Exponential Integral, Up: Exponential Integrals [Index]
These routines compute the exponential integral Ei(x),
Ei(x) := - PV(\int_{-x}^\infty dt \exp(-t)/t)
where PV denotes the principal value of the integral.
Next: Physical Constants, Previous: Sparse BLAS Support, Up: Top [Index]
This chapter describes functions for solving sparse linear systems
of equations. The library provides linear algebra routines which
operate directly on the gsl_spmatrix and gsl_vector
objects.
The functions described in this chapter are declared in the header file gsl_splinalg.h.
| • Overview of Sparse Linear Algebra: | ||
| • Sparse Iterative Solvers: | ||
| • Sparse Linear Algebra Examples: | ||
| • Sparse Linear Algebra References and Further Reading: |
Next: Force and Energy, Previous: Light and Illumination, Up: Physical Constants [Index]
GSL_CONST_MKSA_CURIEThe activity of 1 curie.
GSL_CONST_MKSA_ROENTGENThe exposure of 1 roentgen.
GSL_CONST_MKSA_RADThe absorbed dose of 1 rad.
Next: Iterating the Sparse Linear System, Previous: Sparse Iterative Solver Overview, Up: Sparse Iterative Solvers [Index]
The sparse linear algebra library provides the following types of iterative solvers:
This specifies the Generalized Minimum Residual Method (GMRES). This is a projection method using {\cal K} = {\cal K}_m and {\cal L} = A {\cal K}_m where {\cal K}_m is the m-th Krylov subspace
K_m = span( r_0, A r_0, ..., A^(m-1) r_0)
and r_0 = b - A x_0 is the residual vector of the initial guess x_0. If m is set equal to n, then the Krylov subspace is {\bf R}^n and GMRES will provide the exact solution x. However, the goal is for the method to arrive at a very good approximation to x using a much smaller subspace {\cal K}_m. By default, the GMRES method selects m = MIN(n,10) but the user may specify a different value for m. The GMRES storage requirements grow as O(n(m+1)) and the number of flops grow as O(4 m^2 n - 4 m^3 / 3).
In the below function gsl_splinalg_itersolve_iterate, one
GMRES iteration is defined as projecting the approximate solution
vector onto each Krylov subspace {\cal K}_1, ..., {\cal K}_m,
and so m can be kept smaller by "restarting" the method
and calling gsl_splinalg_itersolve_iterate multiple times,
providing the updated approximation x to each new call. If
the method is not adequately converging, the user may try increasing
the parameter m.
GMRES is considered a robust general purpose iterative solver, however there are cases where the method stagnates if the matrix is not positive-definite and fails to reduce the residual until the very last projection onto the subspace {\cal K}_n = {\bf R}^n. In these cases, preconditioning the linear system can help, but GSL does not currently provide any preconditioners.
Next: Iterating the Sparse Linear System, Previous: Sparse Iterative Solver Overview, Up: Sparse Iterative Solvers [Index]
Next: Support for different numeric types, Previous: Portability functions, Up: Using the library [Index]
The main implementation of some functions in the library will not be optimal on all architectures. For example, there are several ways to compute a Gaussian random variate and their relative speeds are platform-dependent. In cases like this the library provides alternative implementations of these functions with the same interface. If you write your application using calls to the standard implementation you can select an alternative version later via a preprocessor definition. It is also possible to introduce your own optimized functions this way while retaining portability. The following lines demonstrate the use of a platform-dependent choice of methods for sampling from the Gaussian distribution,
#ifdef SPARC #define gsl_ran_gaussian gsl_ran_gaussian_ratio_method #endif #ifdef INTEL #define gsl_ran_gaussian my_gaussian #endif
These lines would be placed in the configuration header file config.h of the application, which should then be included by all the source files. Note that the alternative implementations will not produce bit-for-bit identical results, and in the case of random number distributions will produce an entirely different stream of random variates.
gsl-ref-html-2.3/Gegenbauer-Functions.html 0000664 0001750 0001750 00000014443 13055414530 016641 0 ustar edd eddNext: Hypergeometric Functions, Previous: Gamma and Beta Functions, Up: Special Functions [Index]
The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter 22, where they are known as Ultraspherical polynomials. The functions described in this section are declared in the header file gsl_sf_gegenbauer.h.
These functions evaluate the Gegenbauer polynomials C^{(\lambda)}_n(x) using explicit representations for n =1, 2, 3.
These functions evaluate the Gegenbauer polynomial C^{(\lambda)}_n(x) for a specific value of n, lambda, x subject to \lambda > -1/2, n >= 0.
This function computes an array of Gegenbauer polynomials C^{(\lambda)}_n(x) for n = 0, 1, 2, \dots, nmax, subject to \lambda > -1/2, nmax >= 0.
Next: Ei(x), Up: Exponential Integrals [Index]
These routines compute the exponential integral E_1(x),
E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t.
These routines compute the second-order exponential integral E_2(x),
E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.
These routines compute the exponential integral E_n(x) of order n,
E_n(x) := \Re \int_1^\infty dt \exp(-xt)/t^n.
Next: Nonlinear Least-Squares Geodesic Acceleration Example, Up: Nonlinear Least-Squares Examples [Index]
The following example program fits a weighted exponential model with
background to experimental data, Y = A \exp(-\lambda t) + b. The
first part of the program sets up the functions expb_f and
expb_df to calculate the model and its Jacobian. The appropriate
fitting function is given by,
f_i = (A \exp(-\lambda t_i) + b) - y_i
where we have chosen t_i = i. The Jacobian matrix J is the derivative of these functions with respect to the three parameters (A, \lambda, b). It is given by,
J_{ij} = d f_i / d x_j
where x_0 = A, x_1 = \lambda and x_2 = b. The ith row of the Jacobian is therefore
The main part of the program sets up a Levenberg-Marquardt solver and some simulated random data. The data uses the known parameters (5.0,0.1,1.0) combined with Gaussian noise (standard deviation = 0.1) over a range of 40 timesteps. The initial guess for the parameters is chosen as (1.0, 1.0, 0.0). The iteration terminates when the relative change in x is smaller than 10^{-8}, or when the magnitude of the gradient falls below 10^{-8}. Here are the results of running the program:
iter 0: A = 1.0000, lambda = 1.0000, b = 0.0000, cond(J) = inf, |f(x)| = 62.2029 iter 1: A = 1.2196, lambda = 0.3663, b = 0.0436, cond(J) = 53.6368, |f(x)| = 59.8062 iter 2: A = 1.6062, lambda = 0.1506, b = 0.1054, cond(J) = 23.8178, |f(x)| = 53.9039 iter 3: A = 2.4528, lambda = 0.0583, b = 0.2470, cond(J) = 20.0493, |f(x)| = 28.8039 iter 4: A = 2.9723, lambda = 0.0494, b = 0.3727, cond(J) = 94.5601, |f(x)| = 15.3252 iter 5: A = 3.3473, lambda = 0.0477, b = 0.4410, cond(J) = 229.3627, |f(x)| = 10.7511 iter 6: A = 3.6690, lambda = 0.0508, b = 0.4617, cond(J) = 298.3589, |f(x)| = 9.7373 iter 7: A = 3.9907, lambda = 0.0580, b = 0.5433, cond(J) = 250.0194, |f(x)| = 8.7661 iter 8: A = 4.2353, lambda = 0.0731, b = 0.7989, cond(J) = 154.8571, |f(x)| = 7.4299 iter 9: A = 4.6573, lambda = 0.0958, b = 1.0302, cond(J) = 140.2265, |f(x)| = 6.1893 iter 10: A = 5.0138, lambda = 0.1060, b = 1.0329, cond(J) = 109.4141, |f(x)| = 5.4961 iter 11: A = 5.1505, lambda = 0.1103, b = 1.0497, cond(J) = 100.8762, |f(x)| = 5.4552 iter 12: A = 5.1724, lambda = 0.1110, b = 1.0526, cond(J) = 97.3403, |f(x)| = 5.4542 iter 13: A = 5.1737, lambda = 0.1110, b = 1.0528, cond(J) = 96.7136, |f(x)| = 5.4542 iter 14: A = 5.1738, lambda = 0.1110, b = 1.0528, cond(J) = 96.6678, |f(x)| = 5.4542 iter 15: A = 5.1738, lambda = 0.1110, b = 1.0528, cond(J) = 96.6663, |f(x)| = 5.4542 iter 16: A = 5.1738, lambda = 0.1110, b = 1.0528, cond(J) = 96.6663, |f(x)| = 5.4542 summary from method 'trust-region/levenberg-marquardt' number of iterations: 16 function evaluations: 23 Jacobian evaluations: 17 reason for stopping: small step size initial |f(x)| = 62.202928 final |f(x)| = 5.454180 chisq/dof = 0.804002 A = 5.17379 +/- 0.27938 lambda = 0.11104 +/- 0.00817 b = 1.05283 +/- 0.05365 status = success
The approximate values of the parameters are found correctly, and the chi-squared value indicates a good fit (the chi-squared per degree of freedom is approximately 1). In this case the errors on the parameters can be estimated from the square roots of the diagonal elements of the covariance matrix. If the chi-squared value shows a poor fit (i.e. chi^2/dof >> 1) then the error estimates obtained from the covariance matrix will be too small. In the example program the error estimates are multiplied by \sqrt{\chi^2/dof} in this case, a common way of increasing the errors for a poor fit. Note that a poor fit will result from the use of an inappropriate model, and the scaled error estimates may then be outside the range of validity for Gaussian errors.
Additionally, we see that the condition number of J(x) stays reasonably small throughout the iteration. This indicates we could safely switch to the Cholesky solver for speed improvement, although this particular system is too small to really benefit.
#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_multifit_nlinear.h>
/* number of data points to fit */
#define N 40
struct data {
size_t n;
double * y;
};
int
expb_f (const gsl_vector * x, void *data,
gsl_vector * f)
{
size_t n = ((struct data *)data)->n;
double *y = ((struct data *)data)->y;
double A = gsl_vector_get (x, 0);
double lambda = gsl_vector_get (x, 1);
double b = gsl_vector_get (x, 2);
size_t i;
for (i = 0; i < n; i++)
{
/* Model Yi = A * exp(-lambda * i) + b */
double t = i;
double Yi = A * exp (-lambda * t) + b;
gsl_vector_set (f, i, Yi - y[i]);
}
return GSL_SUCCESS;
}
int
expb_df (const gsl_vector * x, void *data,
gsl_matrix * J)
{
size_t n = ((struct data *)data)->n;
double A = gsl_vector_get (x, 0);
double lambda = gsl_vector_get (x, 1);
size_t i;
for (i = 0; i < n; i++)
{
/* Jacobian matrix J(i,j) = dfi / dxj, */
/* where fi = (Yi - yi)/sigma[i], */
/* Yi = A * exp(-lambda * i) + b */
/* and the xj are the parameters (A,lambda,b) */
double t = i;
double e = exp(-lambda * t);
gsl_matrix_set (J, i, 0, e);
gsl_matrix_set (J, i, 1, -t * A * e);
gsl_matrix_set (J, i, 2, 1.0);
}
return GSL_SUCCESS;
}
void
callback(const size_t iter, void *params,
const gsl_multifit_nlinear_workspace *w)
{
gsl_vector *f = gsl_multifit_nlinear_residual(w);
gsl_vector *x = gsl_multifit_nlinear_position(w);
double rcond;
/* compute reciprocal condition number of J(x) */
gsl_multifit_nlinear_rcond(&rcond, w);
fprintf(stderr, "iter %2zu: A = %.4f, lambda = %.4f, b = %.4f, cond(J) = %8.4f, |f(x)| = %.4f\n",
iter,
gsl_vector_get(x, 0),
gsl_vector_get(x, 1),
gsl_vector_get(x, 2),
1.0 / rcond,
gsl_blas_dnrm2(f));
}
int
main (void)
{
const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
gsl_multifit_nlinear_workspace *w;
gsl_multifit_nlinear_fdf fdf;
gsl_multifit_nlinear_parameters fdf_params =
gsl_multifit_nlinear_default_parameters();
const size_t n = N;
const size_t p = 3;
gsl_vector *f;
gsl_matrix *J;
gsl_matrix *covar = gsl_matrix_alloc (p, p);
double y[N], weights[N];
struct data d = { n, y };
double x_init[3] = { 1.0, 1.0, 0.0 }; /* starting values */
gsl_vector_view x = gsl_vector_view_array (x_init, p);
gsl_vector_view wts = gsl_vector_view_array(weights, n);
gsl_rng * r;
double chisq, chisq0;
int status, info;
size_t i;
const double xtol = 1e-8;
const double gtol = 1e-8;
const double ftol = 0.0;
gsl_rng_env_setup();
r = gsl_rng_alloc(gsl_rng_default);
/* define the function to be minimized */
fdf.f = expb_f;
fdf.df = expb_df; /* set to NULL for finite-difference Jacobian */
fdf.fvv = NULL; /* not using geodesic acceleration */
fdf.n = n;
fdf.p = p;
fdf.params = &d;
/* this is the data to be fitted */
for (i = 0; i < n; i++)
{
double t = i;
double yi = 1.0 + 5 * exp (-0.1 * t);
double si = 0.1 * yi;
double dy = gsl_ran_gaussian(r, si);
weights[i] = 1.0 / (si * si);
y[i] = yi + dy;
printf ("data: %zu %g %g\n", i, y[i], si);
};
/* allocate workspace with default parameters */
w = gsl_multifit_nlinear_alloc (T, &fdf_params, n, p);
/* initialize solver with starting point and weights */
gsl_multifit_nlinear_winit (&x.vector, &wts.vector, &fdf, w);
/* compute initial cost function */
f = gsl_multifit_nlinear_residual(w);
gsl_blas_ddot(f, f, &chisq0);
/* solve the system with a maximum of 20 iterations */
status = gsl_multifit_nlinear_driver(20, xtol, gtol, ftol,
callback, NULL, &info, w);
/* compute covariance of best fit parameters */
J = gsl_multifit_nlinear_jac(w);
gsl_multifit_nlinear_covar (J, 0.0, covar);
/* compute final cost */
gsl_blas_ddot(f, f, &chisq);
#define FIT(i) gsl_vector_get(w->x, i)
#define ERR(i) sqrt(gsl_matrix_get(covar,i,i))
fprintf(stderr, "summary from method '%s/%s'\n",
gsl_multifit_nlinear_name(w),
gsl_multifit_nlinear_trs_name(w));
fprintf(stderr, "number of iterations: %zu\n",
gsl_multifit_nlinear_niter(w));
fprintf(stderr, "function evaluations: %zu\n", fdf.nevalf);
fprintf(stderr, "Jacobian evaluations: %zu\n", fdf.nevaldf);
fprintf(stderr, "reason for stopping: %s\n",
(info == 1) ? "small step size" : "small gradient");
fprintf(stderr, "initial |f(x)| = %f\n", sqrt(chisq0));
fprintf(stderr, "final |f(x)| = %f\n", sqrt(chisq));
{
double dof = n - p;
double c = GSL_MAX_DBL(1, sqrt(chisq / dof));
fprintf(stderr, "chisq/dof = %g\n", chisq / dof);
fprintf (stderr, "A = %.5f +/- %.5f\n", FIT(0), c*ERR(0));
fprintf (stderr, "lambda = %.5f +/- %.5f\n", FIT(1), c*ERR(1));
fprintf (stderr, "b = %.5f +/- %.5f\n", FIT(2), c*ERR(2));
}
fprintf (stderr, "status = %s\n", gsl_strerror (status));
gsl_multifit_nlinear_free (w);
gsl_matrix_free (covar);
gsl_rng_free (r);
return 0;
}
Next: Nonlinear Least-Squares Geodesic Acceleration Example, Up: Nonlinear Least-Squares Examples [Index]
Next: Eigensystems, Previous: BLAS Support, Up: Top [Index]
This chapter describes functions for solving linear systems. The
library provides linear algebra operations which operate directly on
the gsl_vector and gsl_matrix objects. These routines
use the standard algorithms from Golub & Van Loan’s Matrix
Computations with Level-1 and Level-2 BLAS calls for efficiency.
The functions described in this chapter are declared in the header file gsl_linalg.h.
Next: Eigensystems, Previous: BLAS Support, Up: Top [Index]
Next: Multimin References and Further Reading, Previous: Multimin Algorithms without Derivatives, Up: Multidimensional Minimization [Index]
This example program finds the minimum of the paraboloid function defined earlier. The location of the minimum is offset from the origin in x and y, and the function value at the minimum is non-zero. The main program is given below, it requires the example function given earlier in this chapter.
int
main (void)
{
size_t iter = 0;
int status;
const gsl_multimin_fdfminimizer_type *T;
gsl_multimin_fdfminimizer *s;
/* Position of the minimum (1,2), scale factors
10,20, height 30. */
double par[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 };
gsl_vector *x;
gsl_multimin_function_fdf my_func;
my_func.n = 2;
my_func.f = my_f;
my_func.df = my_df;
my_func.fdf = my_fdf;
my_func.params = par;
/* Starting point, x = (5,7) */
x = gsl_vector_alloc (2);
gsl_vector_set (x, 0, 5.0);
gsl_vector_set (x, 1, 7.0);
T = gsl_multimin_fdfminimizer_conjugate_fr;
s = gsl_multimin_fdfminimizer_alloc (T, 2);
gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4);
do
{
iter++;
status = gsl_multimin_fdfminimizer_iterate (s);
if (status)
break;
status = gsl_multimin_test_gradient (s->gradient, 1e-3);
if (status == GSL_SUCCESS)
printf ("Minimum found at:\n");
printf ("%5d %.5f %.5f %10.5f\n", iter,
gsl_vector_get (s->x, 0),
gsl_vector_get (s->x, 1),
s->f);
}
while (status == GSL_CONTINUE && iter < 100);
gsl_multimin_fdfminimizer_free (s);
gsl_vector_free (x);
return 0;
}
The initial step-size is chosen as 0.01, a conservative estimate in this case, and the line minimization parameter is set at 0.0001. The program terminates when the norm of the gradient has been reduced below 0.001. The output of the program is shown below,
x y f
1 4.99629 6.99072 687.84780
2 4.98886 6.97215 683.55456
3 4.97400 6.93501 675.01278
4 4.94429 6.86073 658.10798
5 4.88487 6.71217 625.01340
6 4.76602 6.41506 561.68440
7 4.52833 5.82083 446.46694
8 4.05295 4.63238 261.79422
9 3.10219 2.25548 75.49762
10 2.85185 1.62963 67.03704
11 2.19088 1.76182 45.31640
12 0.86892 2.02622 30.18555
Minimum found at:
13 1.00000 2.00000 30.00000
Note that the algorithm gradually increases the step size as it successfully moves downhill, as can be seen by plotting the successive points.
The conjugate gradient algorithm finds the minimum on its second direction because the function is purely quadratic. Additional iterations would be needed for a more complicated function.
Here is another example using the Nelder-Mead Simplex algorithm to minimize the same example object function, as above.
int
main(void)
{
double par[5] = {1.0, 2.0, 10.0, 20.0, 30.0};
const gsl_multimin_fminimizer_type *T =
gsl_multimin_fminimizer_nmsimplex2;
gsl_multimin_fminimizer *s = NULL;
gsl_vector *ss, *x;
gsl_multimin_function minex_func;
size_t iter = 0;
int status;
double size;
/* Starting point */
x = gsl_vector_alloc (2);
gsl_vector_set (x, 0, 5.0);
gsl_vector_set (x, 1, 7.0);
/* Set initial step sizes to 1 */
ss = gsl_vector_alloc (2);
gsl_vector_set_all (ss, 1.0);
/* Initialize method and iterate */
minex_func.n = 2;
minex_func.f = my_f;
minex_func.params = par;
s = gsl_multimin_fminimizer_alloc (T, 2);
gsl_multimin_fminimizer_set (s, &minex_func, x, ss);
do
{
iter++;
status = gsl_multimin_fminimizer_iterate(s);
if (status)
break;
size = gsl_multimin_fminimizer_size (s);
status = gsl_multimin_test_size (size, 1e-2);
if (status == GSL_SUCCESS)
{
printf ("converged to minimum at\n");
}
printf ("%5d %10.3e %10.3e f() = %7.3f size = %.3f\n",
iter,
gsl_vector_get (s->x, 0),
gsl_vector_get (s->x, 1),
s->fval, size);
}
while (status == GSL_CONTINUE && iter < 100);
gsl_vector_free(x);
gsl_vector_free(ss);
gsl_multimin_fminimizer_free (s);
return status;
}
The minimum search stops when the Simplex size drops to 0.01. The output is shown below.
1 6.500e+00 5.000e+00 f() = 512.500 size = 1.130
2 5.250e+00 4.000e+00 f() = 290.625 size = 1.409
3 5.250e+00 4.000e+00 f() = 290.625 size = 1.409
4 5.500e+00 1.000e+00 f() = 252.500 size = 1.409
5 2.625e+00 3.500e+00 f() = 101.406 size = 1.847
6 2.625e+00 3.500e+00 f() = 101.406 size = 1.847
7 0.000e+00 3.000e+00 f() = 60.000 size = 1.847
8 2.094e+00 1.875e+00 f() = 42.275 size = 1.321
9 2.578e-01 1.906e+00 f() = 35.684 size = 1.069
10 5.879e-01 2.445e+00 f() = 35.664 size = 0.841
11 1.258e+00 2.025e+00 f() = 30.680 size = 0.476
12 1.258e+00 2.025e+00 f() = 30.680 size = 0.367
13 1.093e+00 1.849e+00 f() = 30.539 size = 0.300
14 8.830e-01 2.004e+00 f() = 30.137 size = 0.172
15 8.830e-01 2.004e+00 f() = 30.137 size = 0.126
16 9.582e-01 2.060e+00 f() = 30.090 size = 0.106
17 1.022e+00 2.004e+00 f() = 30.005 size = 0.063
18 1.022e+00 2.004e+00 f() = 30.005 size = 0.043
19 1.022e+00 2.004e+00 f() = 30.005 size = 0.043
20 1.022e+00 2.004e+00 f() = 30.005 size = 0.027
21 1.022e+00 2.004e+00 f() = 30.005 size = 0.022
22 9.920e-01 1.997e+00 f() = 30.001 size = 0.016
23 9.920e-01 1.997e+00 f() = 30.001 size = 0.013
converged to minimum at
24 9.920e-01 1.997e+00 f() = 30.001 size = 0.008
The simplex size first increases, while the simplex moves towards the minimum. After a while the size begins to decrease as the simplex contracts around the minimum.
Next: Multimin References and Further Reading, Previous: Multimin Algorithms without Derivatives, Up: Multidimensional Minimization [Index]
Next: Properties of complex numbers, Up: Complex Numbers [Index]
Complex numbers are represented using the type gsl_complex. The
internal representation of this type may vary across platforms and
should not be accessed directly. The functions and macros described
below allow complex numbers to be manipulated in a portable way.
For reference, the default form of the gsl_complex type is
given by the following struct,
typedef struct
{
double dat[2];
} gsl_complex;
The real and imaginary part are stored in contiguous elements of a two
element array. This eliminates any padding between the real and
imaginary parts, dat[0] and dat[1], allowing the struct to
be mapped correctly onto packed complex arrays.
This function uses the rectangular Cartesian components
(x,y) to return the complex number z = x + i y. An inline version of this function is used when HAVE_INLINE is defined.
This function returns the complex number z = r \exp(i \theta) = r (\cos(\theta) + i \sin(\theta)) from the polar representation (r,theta).
These macros return the real and imaginary parts of the complex number z.
This macro uses the Cartesian components (x,y) to set the real and imaginary parts of the complex number pointed to by zp. For example,
GSL_SET_COMPLEX(&z, 3, 4)
sets z to be 3 + 4i.
These macros allow the real and imaginary parts of the complex number pointed to by zp to be set independently.
Next: Properties of complex numbers, Up: Complex Numbers [Index]
Next: Driver, Previous: Adaptive Step-size Control, Up: Ordinary Differential Equations [Index]
The evolution function combines the results of a stepping function and control function to reliably advance the solution forward one step using an acceptable step-size.
This function returns a pointer to a newly allocated instance of an evolution function for a system of dim dimensions.
This function advances the system (e, sys) from time t and position y using the stepping function step. The new time and position are stored in t and y on output.
The initial step-size is taken as h. The control function
con is applied to check whether the local error estimated by the
stepping function step using step-size h exceeds the
required error tolerance. If the error is too high, the step is
retried by calling step with a decreased step-size. This process
is continued until an acceptable step-size is found. An estimate of
the local error for the step can be obtained from the components of
the array e->yerr[].
If the user-supplied functions defined in the system sys returns
GSL_EBADFUNC, the function returns immediately with the same
return code. In this case the user must call
gsl_odeiv2_step_reset and
gsl_odeiv2_evolve_reset before calling this function again.
Otherwise, if the user-supplied functions defined in the system
sys or the stepping function step return a status other
than GSL_SUCCESS, the step is retried with a decreased
step-size. If the step-size decreases below machine precision, a
status of GSL_FAILURE is returned if the user functions
returned GSL_SUCCESS. Otherwise the value returned by user
function is returned. If no acceptable step can be made, t and
y will be restored to their pre-step values and h contains
the final attempted step-size.
If the step is successful the function returns a suggested step-size for the next step in h. The maximum time t1 is guaranteed not to be exceeded by the time-step. On the final time-step the value of t will be set to t1 exactly.
This function advances the ODE-system (e, sys, con)
from time t and position y using the stepping function
step by a specified step size h. If the local error
estimated by the stepping function exceeds the desired error level,
the step is not taken and the function returns
GSL_FAILURE. Otherwise the value returned by user function is
returned.
This function resets the evolution function e. It should be used whenever the next use of e will not be a continuation of a previous step.
This function frees all the memory associated with the evolution function e.
This function sets a pointer of the driver object d for evolve object e.
If a system has discontinuous changes in the derivatives at known points, it is advisable to evolve the system between each discontinuity in sequence. For example, if a step-change in an external driving force occurs at times t_a, t_b and t_c then evolution should be carried out over the ranges (t_0,t_a), (t_a,t_b), (t_b,t_c), and (t_c,t_1) separately and not directly over the range (t_0,t_1).
Next: Driver, Previous: Adaptive Step-size Control, Up: Ordinary Differential Equations [Index]
Next: Psi (Digamma) Function, Previous: Mathieu Functions, Up: Special Functions [Index]
The following functions are equivalent to the function gsl_pow_int
(see Small integer powers) with an error estimate. These functions are
declared in the header file gsl_sf_pow_int.h.
These routines compute the power x^n for integer n. The power is computed using the minimum number of multiplications. For example, x^8 is computed as ((x^2)^2)^2, requiring only 3 multiplications. For reasons of efficiency, these functions do not check for overflow or underflow conditions.
#include <gsl/gsl_sf_pow_int.h> /* compute 3.0**12 */ double y = gsl_sf_pow_int(3.0, 12);
Next: Root Finding Examples, Previous: Root Bracketing Algorithms, Up: One dimensional Root-Finding [Index]
The root polishing algorithms described in this section require an initial guess for the location of the root. There is no absolute guarantee of convergence—the function must be suitable for this technique and the initial guess must be sufficiently close to the root for it to work. When these conditions are satisfied then convergence is quadratic.
These algorithms make use of both the function and its derivative.
Newton’s Method is the standard root-polishing algorithm. The algorithm begins with an initial guess for the location of the root. On each iteration, a line tangent to the function f is drawn at that position. The point where this line crosses the x-axis becomes the new guess. The iteration is defined by the following sequence,
x_{i+1} = x_i - f(x_i)/f'(x_i)
Newton’s method converges quadratically for single roots, and linearly for multiple roots.
The secant method is a simplified version of Newton’s method which does not require the computation of the derivative on every step.
On its first iteration the algorithm begins with Newton’s method, using the derivative to compute a first step,
x_1 = x_0 - f(x_0)/f'(x_0)
Subsequent iterations avoid the evaluation of the derivative by replacing it with a numerical estimate, the slope of the line through the previous two points,
x_{i+1} = x_i f(x_i) / f'_{est} where
f'_{est} = (f(x_i) - f(x_{i-1})/(x_i - x_{i-1})
When the derivative does not change significantly in the vicinity of the root the secant method gives a useful saving. Asymptotically the secant method is faster than Newton’s method whenever the cost of evaluating the derivative is more than 0.44 times the cost of evaluating the function itself. As with all methods of computing a numerical derivative the estimate can suffer from cancellation errors if the separation of the points becomes too small.
On single roots, the method has a convergence of order (1 + \sqrt 5)/2 (approximately 1.62). It converges linearly for multiple roots.
The Steffenson Method14 provides the fastest convergence of all the routines. It combines the basic Newton algorithm with an Aitken “delta-squared” acceleration. If the Newton iterates are x_i then the acceleration procedure generates a new sequence R_i,
R_i = x_i - (x_{i+1} - x_i)^2 / (x_{i+2} - 2 x_{i+1} + x_{i})
which converges faster than the original sequence under reasonable conditions. The new sequence requires three terms before it can produce its first value so the method returns accelerated values on the second and subsequent iterations. On the first iteration it returns the ordinary Newton estimate. The Newton iterate is also returned if the denominator of the acceleration term ever becomes zero.
As with all acceleration procedures this method can become unstable if the function is not well-behaved.
J.F. Steffensen (1873–1961). The spelling used in the name of the function is slightly incorrect, but has been preserved to avoid incompatibility.
Next: Root Finding Examples, Previous: Root Bracketing Algorithms, Up: One dimensional Root-Finding [Index]
The following program computes a least squares straight-line fit to a simple dataset, and outputs the best-fit line and its associated one standard-deviation error bars.
#include <stdio.h>
#include <gsl/gsl_fit.h>
int
main (void)
{
int i, n = 4;
double x[4] = { 1970, 1980, 1990, 2000 };
double y[4] = { 12, 11, 14, 13 };
double w[4] = { 0.1, 0.2, 0.3, 0.4 };
double c0, c1, cov00, cov01, cov11, chisq;
gsl_fit_wlinear (x, 1, w, 1, y, 1, n,
&c0, &c1, &cov00, &cov01, &cov11,
&chisq);
printf ("# best fit: Y = %g + %g X\n", c0, c1);
printf ("# covariance matrix:\n");
printf ("# [ %g, %g\n# %g, %g]\n",
cov00, cov01, cov01, cov11);
printf ("# chisq = %g\n", chisq);
for (i = 0; i < n; i++)
printf ("data: %g %g %g\n",
x[i], y[i], 1/sqrt(w[i]));
printf ("\n");
for (i = -30; i < 130; i++)
{
double xf = x[0] + (i/100.0) * (x[n-1] - x[0]);
double yf, yf_err;
gsl_fit_linear_est (xf,
c0, c1,
cov00, cov01, cov11,
&yf, &yf_err);
printf ("fit: %g %g\n", xf, yf);
printf ("hi : %g %g\n", xf, yf + yf_err);
printf ("lo : %g %g\n", xf, yf - yf_err);
}
return 0;
}
The following commands extract the data from the output of the program
and display it using the GNU plotutils graph utility,
$ ./demo > tmp
$ more tmp
# best fit: Y = -106.6 + 0.06 X
# covariance matrix:
# [ 39602, -19.9
# -19.9, 0.01]
# chisq = 0.8
$ for n in data fit hi lo ;
do
grep "^$n" tmp | cut -d: -f2 > $n ;
done
$ graph -T X -X x -Y y -y 0 20 -m 0 -S 2 -Ie data
-S 0 -I a -m 1 fit -m 2 hi -m 2 lo
Next: Random Number Distribution Examples, Previous: The Logarithmic Distribution, Up: Random Number Distributions [Index]
The following functions allow the shuffling and sampling of a set of objects. The algorithms rely on a random number generator as a source of randomness and a poor quality generator can lead to correlations in the output. In particular it is important to avoid generators with a short period. For more information see Knuth, v2, 3rd ed, Section 3.4.2, “Random Sampling and Shuffling”.
This function randomly shuffles the order of n objects, each of size size, stored in the array base[0..n-1]. The output of the random number generator r is used to produce the permutation. The algorithm generates all possible n! permutations with equal probability, assuming a perfect source of random numbers.
The following code shows how to shuffle the numbers from 0 to 51,
int a[52];
for (i = 0; i < 52; i++)
{
a[i] = i;
}
gsl_ran_shuffle (r, a, 52, sizeof (int));
This function fills the array dest[k] with k objects taken randomly from the n elements of the array src[0..n-1]. The objects are each of size size. The output of the random number generator r is used to make the selection. The algorithm ensures all possible samples are equally likely, assuming a perfect source of randomness.
The objects are sampled without replacement, thus each object can
only appear once in dest[k]. It is required that k be less
than or equal to n. The objects in dest will be in the
same relative order as those in src. You will need to call
gsl_ran_shuffle(r, dest, n, size) if you want to randomize the
order.
The following code shows how to select a random sample of three unique numbers from the set 0 to 99,
double a[3], b[100];
for (i = 0; i < 100; i++)
{
b[i] = (double) i;
}
gsl_ran_choose (r, a, 3, b, 100, sizeof (double));
This function is like gsl_ran_choose but samples k items
from the original array of n items src with replacement, so
the same object can appear more than once in the output sequence
dest. There is no requirement that k be less than n
in this case.
Next: Random Number Distribution Examples, Previous: The Logarithmic Distribution, Up: Random Number Distributions [Index]
Next: Searching histogram ranges, Previous: Copying Histograms, Up: Histograms [Index]
There are two ways to access histogram bins, either by specifying an x coordinate or by using the bin-index directly. The functions for accessing the histogram through x coordinates use a binary search to identify the bin which covers the appropriate range.
This function updates the histogram h by adding one (1.0) to the bin whose range contains the coordinate x.
If x lies in the valid range of the histogram then the function
returns zero to indicate success. If x is less than the lower
limit of the histogram then the function returns GSL_EDOM, and
none of bins are modified. Similarly, if the value of x is greater
than or equal to the upper limit of the histogram then the function
returns GSL_EDOM, and none of the bins are modified. The error
handler is not called, however, since it is often necessary to compute
histograms for a small range of a larger dataset, ignoring the values
outside the range of interest.
This function is similar to gsl_histogram_increment but increases
the value of the appropriate bin in the histogram h by the
floating-point number weight.
This function returns the contents of the i-th bin of the histogram
h. If i lies outside the valid range of indices for the
histogram then the error handler is called with an error code of
GSL_EDOM and the function returns 0.
This function finds the upper and lower range limits of the i-th
bin of the histogram h. If the index i is valid then the
corresponding range limits are stored in lower and upper.
The lower limit is inclusive (i.e. events with this coordinate are
included in the bin) and the upper limit is exclusive (i.e. events with
the coordinate of the upper limit are excluded and fall in the
neighboring higher bin, if it exists). The function returns 0 to
indicate success. If i lies outside the valid range of indices for
the histogram then the error handler is called and the function returns
an error code of GSL_EDOM.
These functions return the maximum upper and minimum lower range limits
and the number of bins of the histogram h. They provide a way of
determining these values without accessing the gsl_histogram
struct directly.
This function resets all the bins in the histogram h to zero.
Next: Searching histogram ranges, Previous: Copying Histograms, Up: Histograms [Index]
Next: Sparse Matrices, Previous: Nonlinear Least-Squares Fitting, Up: Top [Index]
This chapter describes functions for the computation of smoothing basis splines (B-splines). A smoothing spline differs from an interpolating spline in that the resulting curve is not required to pass through each datapoint. See Interpolation, for information about interpolating splines.
The header file gsl_bspline.h contains the prototypes for the bspline functions and related declarations.
Next: Large Dense Linear Systems, Previous: Regularized regression, Up: Least-Squares Fitting [Index]
Ordinary least squares (OLS) models are often heavily influenced by the presence of outliers. Outliers are data points which do not follow the general trend of the other observations, although there is strictly no precise definition of an outlier. Robust linear regression refers to regression algorithms which are robust to outliers. The most common type of robust regression is M-estimation. The general M-estimator minimizes the objective function
\sum_i \rho(e_i) = \sum_i \rho (y_i - Y(c, x_i))
where e_i = y_i - Y(c, x_i) is the residual of the ith data point, and \rho(e_i) is a function which should have the following properties:
The special case of ordinary least squares is given by \rho(e_i) = e_i^2. Letting \psi = \rho' be the derivative of \rho, differentiating the objective function with respect to the coefficients c and setting the partial derivatives to zero produces the system of equations
\sum_i \psi(e_i) X_i = 0
where X_i is a vector containing row i of the design matrix X. Next, we define a weight function w(e) = \psi(e)/e, and let w_i = w(e_i):
\sum_i w_i e_i X_i = 0
This system of equations is equivalent to solving a weighted ordinary least squares problem, minimizing \chi^2 = \sum_i w_i e_i^2. The weights however, depend on the residuals e_i, which depend on the coefficients c, which depend on the weights. Therefore, an iterative solution is used, called Iteratively Reweighted Least Squares (IRLS).
|c_i^(k) - c_i^(k-1)| \le \epsilon \times max(|c_i^(k)|, |c_i^(k-1)|)
for all 0 \le i < p where \epsilon is a small tolerance factor.
The key to this method lies in selecting the function \psi(e_i) to assign smaller weights to large residuals, and larger weights to smaller residuals. As the iteration proceeds, outliers are assigned smaller and smaller weights, eventually having very little or no effect on the fitted model.
This function allocates a workspace for fitting a model to n observations using p parameters. The size of the workspace is O(np + p^2). The type T specifies the function \psi and can be selected from the following choices.
This specifies the gsl_multifit_robust_bisquare type (see below) and is a good
general purpose choice for robust regression.
This is Tukey’s biweight (bisquare) function and is a good general purpose choice for robust regression. The weight function is given by
w(e) = (1 - e^2)^2
and the default tuning constant is t = 4.685.
This is Cauchy’s function, also known as the Lorentzian function. This function does not guarantee a unique solution, meaning different choices of the coefficient vector c could minimize the objective function. Therefore this option should be used with care. The weight function is given by
w(e) = 1 / (1 + e^2)
and the default tuning constant is t = 2.385.
This is the fair \rho function, which guarantees a unique solution and has continuous derivatives to three orders. The weight function is given by
w(e) = 1 / (1 + |e|)
and the default tuning constant is t = 1.400.
This specifies Huber’s \rho function, which is a parabola in the vicinity of zero and increases linearly for a given threshold |e| > t. This function is also considered an excellent general purpose robust estimator, however, occasional difficulties can be encountered due to the discontinuous first derivative of the \psi function. The weight function is given by
w(e) = 1/max(1,|e|)
and the default tuning constant is t = 1.345.
This function frees the memory associated with the workspace w.
This function returns the name of the robust type T specified to gsl_multifit_robust_alloc.
This function sets the tuning constant t used to adjust the residuals at each iteration to tune. Decreasing the tuning constant increases the downweight assigned to large residuals, while increasing the tuning constant decreases the downweight assigned to large residuals.
This function sets the maximum number of iterations in the iteratively
reweighted least squares algorithm to maxiter. By default,
this value is set to 100 by gsl_multifit_robust_alloc.
This function assigns weights to the vector wts using the residual vector r and
previously specified weighting function. The output weights are given by wts_i = w(r_i / (t \sigma)),
where the weighting functions w are detailed in gsl_multifit_robust_alloc. \sigma
is an estimate of the residual standard deviation based on the Median-Absolute-Deviation and t
is the tuning constant. This
function is useful if the user wishes to implement their own robust regression rather than using
the supplied gsl_multifit_robust routine below.
This function computes the best-fit parameters c of the model
y = X c for the observations y and the matrix of
predictor variables X, attemping to reduce the influence
of outliers using the algorithm outlined above.
The p-by-p variance-covariance matrix of the model parameters
cov is estimated as \sigma^2 (X^T X)^{-1}, where \sigma is
an approximation of the residual standard deviation using the theory of robust
regression. Special care must be taken when estimating \sigma and
other statistics such as R^2, and so these
are computed internally and are available by calling the function
gsl_multifit_robust_statistics.
If the coefficients do not converge within the maximum iteration
limit, the function returns GSL_EMAXITER. In this case,
the current estimates of the coefficients and covariance matrix
are returned in c and cov and the internal fit statistics
are computed with these estimates.
This function uses the best-fit robust regression coefficients c and their covariance matrix cov to compute the fitted function value y and its standard deviation y_err for the model y = x.c at the point x.
This function computes the vector of studentized residuals
r_i = {y_i - (X c)_i \over \sigma \sqrt{1 - h_i}} for
the observations y, coefficients c and matrix of predictor
variables X. The routine gsl_multifit_robust must
first be called to compute the statisical leverages h_i of
the matrix X and residual standard deviation estimate \sigma.
This function returns a structure containing relevant statistics from a robust regression. The function
gsl_multifit_robust must be called first to perform the regression and calculate these statistics.
The returned gsl_multifit_robust_stats structure contains the following fields.
sigma_ols This contains the standard deviation of the residuals as computed from ordinary least squares (OLS).
sigma_mad This contains an estimate of the standard deviation of the final residuals using the Median-Absolute-Deviation statistic
sigma_rob This contains an estimate of the standard deviation of the final residuals from the theory of robust regression (see Street et al, 1988).
sigma This contains an estimate of the standard deviation of the final residuals by attemping to reconcile sigma_rob and sigma_ols
in a reasonable way.
Rsq This contains the R^2 coefficient of determination statistic using the estimate sigma.
adj_Rsq This contains the adjusted R^2 coefficient of determination statistic using the estimate sigma.
rmse This contains the root mean squared error of the final residuals
sse This contains the residual sum of squares taking into account the robust covariance matrix.
dof This contains the number of degrees of freedom n - p
numit Upon successful convergence, this contains the number of iterations performed
weights This contains the final weight vector of length n
r This contains the final residual vector of length n, r = y - X c
Next: Large Dense Linear Systems, Previous: Regularized regression, Up: Least-Squares Fitting [Index]
Next: The Geometric Distribution, Previous: The Negative Binomial Distribution, Up: Random Number Distributions [Index]
This function returns a random integer from the Pascal distribution. The Pascal distribution is simply a negative binomial distribution with an integer value of n.
p(k) = {(n + k - 1)! \over k! (n - 1)! } p^n (1-p)^k
for k >= 0
This function computes the probability p(k) of obtaining k from a Pascal distribution with parameters p and n, using the formula given above.
These functions compute the cumulative distribution functions P(k), Q(k) for the Pascal distribution with parameters p and n.
Next: Quasi-Random Sequences, Previous: Numerical Integration, Up: Top [Index]
The library provides a large collection of random number generators which can be accessed through a uniform interface. Environment variables allow you to select different generators and seeds at runtime, so that you can easily switch between generators without needing to recompile your program. Each instance of a generator keeps track of its own state, allowing the generators to be used in multi-threaded programs. Additional functions are available for transforming uniform random numbers into samples from continuous or discrete probability distributions such as the Gaussian, log-normal or Poisson distributions.
These functions are declared in the header file gsl_rng.h.
Next: Quasi-Random Sequences, Previous: Numerical Integration, Up: Top [Index]
Next: Reading and writing 2D histograms, Previous: 2D Histogram Statistics, Up: Histograms [Index]
This function returns 1 if all the individual bin ranges of the two histograms are identical, and 0 otherwise.
This function adds the contents of the bins in histogram h2 to the corresponding bins of histogram h1, i.e. h'_1(i,j) = h_1(i,j) + h_2(i,j). The two histograms must have identical bin ranges.
This function subtracts the contents of the bins in histogram h2 from the corresponding bins of histogram h1, i.e. h'_1(i,j) = h_1(i,j) - h_2(i,j). The two histograms must have identical bin ranges.
This function multiplies the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h'_1(i,j) = h_1(i,j) * h_2(i,j). The two histograms must have identical bin ranges.
This function divides the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h'_1(i,j) = h_1(i,j) / h_2(i,j). The two histograms must have identical bin ranges.
This function multiplies the contents of the bins of histogram h by the constant scale, i.e. h'_1(i,j) = h_1(i,j) scale.
This function shifts the contents of the bins of histogram h by the constant offset, i.e. h'_1(i,j) = h_1(i,j) + offset.
Next: Evolution, Previous: Stepping Functions, Up: Ordinary Differential Equations [Index]
The control function examines the proposed change to the solution produced by a stepping function and attempts to determine the optimal step-size for a user-specified level of error.
The standard control object is a four parameter heuristic based on absolute and relative errors eps_abs and eps_rel, and scaling factors a_y and a_dydt for the system state y(t) and derivatives y'(t) respectively.
The step-size adjustment procedure for this method begins by computing the desired error level D_i for each component,
D_i = eps_abs + eps_rel * (a_y |y_i| + a_dydt h |y\prime_i|)
and comparing it with the observed error E_i = |yerr_i|. If the observed error E exceeds the desired error level D by more than 10% for any component then the method reduces the step-size by an appropriate factor,
h_new = h_old * S * (E/D)^(-1/q)
where q is the consistency order of the method (e.g. q=4 for 4(5) embedded RK), and S is a safety factor of 0.9. The ratio E/D is taken to be the maximum of the ratios E_i/D_i.
If the observed error E is less than 50% of the desired error level D for the maximum ratio E_i/D_i then the algorithm takes the opportunity to increase the step-size to bring the error in line with the desired level,
h_new = h_old * S * (E/D)^(-1/(q+1))
This encompasses all the standard error scaling methods. To avoid uncontrolled changes in the stepsize, the overall scaling factor is limited to the range 1/5 to 5.
This function creates a new control object which will keep the local error on each step within an absolute error of eps_abs and relative error of eps_rel with respect to the solution y_i(t). This is equivalent to the standard control object with a_y=1 and a_dydt=0.
This function creates a new control object which will keep the local error on each step within an absolute error of eps_abs and relative error of eps_rel with respect to the derivatives of the solution y'_i(t). This is equivalent to the standard control object with a_y=0 and a_dydt=1.
This function creates a new control object which uses the same algorithm
as gsl_odeiv2_control_standard_new but with an absolute error
which is scaled for each component by the array scale_abs.
The formula for D_i for this control object is,
D_i = eps_abs * s_i + eps_rel * (a_y |y_i| + a_dydt h |y\prime_i|)
where s_i is the i-th component of the array scale_abs. The same error control heuristic is used by the Matlab ODE suite.
This function returns a pointer to a newly allocated instance of a control function of type T. This function is only needed for defining new types of control functions. For most purposes the standard control functions described above should be sufficient.
This function initializes the control function c with the parameters eps_abs (absolute error), eps_rel (relative error), a_y (scaling factor for y) and a_dydt (scaling factor for derivatives).
This function frees all the memory associated with the control function c.
This function adjusts the step-size h using the control function
c, and the current values of y, yerr and dydt.
The stepping function step is also needed to determine the order
of the method. If the error in the y-values yerr is found to be
too large then the step-size h is reduced and the function returns
GSL_ODEIV_HADJ_DEC. If the error is sufficiently small then
h may be increased and GSL_ODEIV_HADJ_INC is returned. The
function returns GSL_ODEIV_HADJ_NIL if the step-size is
unchanged. The goal of the function is to estimate the largest
step-size which satisfies the user-specified accuracy requirements for
the current point.
This function returns a pointer to the name of the control function. For example,
printf ("control method is '%s'\n",
gsl_odeiv2_control_name (c));
would print something like control method is 'standard'
This function calculates the desired error level of the ind-th component to errlev. It requires the value (y) and value of the derivative (dydt) of the component, and the current step size h.
This function sets a pointer of the driver object d for control object c.
Next: Evolution, Previous: Stepping Functions, Up: Ordinary Differential Equations [Index]
The normal equations approach to the large linear least squares problem described above is popular due to its speed and simplicity. Since the normal equations solution to the problem is given by
c = ( X^T X + \lambda^2 I )^-1 X^T y
only the p-by-p matrix X^T X and p-by-1 vector X^T y need to be stored. Using the partition scheme described above, these are given by
X^T X = \sum_i X_i^T X_i X^T y = \sum_i X_i^T y_i
Since the matrix X^T X is symmetric, only half of it needs to be calculated. Once all of the blocks (X_i,y_i) have been accumulated into the final X^T X and X^T y, the system can be solved with a Cholesky factorization of the X^T X matrix. If the Cholesky factorization fails (occasionally due to numerical rounding errors), a QR decomposition is then used. In both cases, the X^T X matrix is first transformed via a diagonal scaling transformation to attempt to reduce its condition number as much as possible to recover a more accurate solution vector. The normal equations approach is the fastest method for solving the large least squares problem, and is accurate for well-conditioned matrices X. However, for ill-conditioned matrices, as is often the case for large systems, this method can suffer from numerical instabilities (see Trefethen and Bau, 1997). The number of operations for this method is O(np^2 + {1 \over 3}p^3).
gsl-ref-html-2.3/Random-Number-Generator-Performance.html 0000664 0001750 0001750 00000011266 13055414571 021465 0 ustar edd eddNext: Random Number Generator Examples, Previous: Other random number generators, Up: Random Number Generation [Index]
The following table shows the relative performance of a selection the
available random number generators. The fastest simulation quality
generators are taus, gfsr4 and mt19937. The
generators which offer the best mathematically-proven quality are those
based on the RANLUX algorithm.
1754 k ints/sec, 870 k doubles/sec, taus 1613 k ints/sec, 855 k doubles/sec, gfsr4 1370 k ints/sec, 769 k doubles/sec, mt19937 565 k ints/sec, 571 k doubles/sec, ranlxs0 400 k ints/sec, 405 k doubles/sec, ranlxs1 490 k ints/sec, 389 k doubles/sec, mrg 407 k ints/sec, 297 k doubles/sec, ranlux 243 k ints/sec, 254 k doubles/sec, ranlxd1 251 k ints/sec, 253 k doubles/sec, ranlxs2 238 k ints/sec, 215 k doubles/sec, cmrg 247 k ints/sec, 198 k doubles/sec, ranlux389 141 k ints/sec, 140 k doubles/sec, ranlxd2
Next: Special Function Modes, Previous: Special Function Usage, Up: Special Functions [Index]
The error handling form of the special functions always calculate an error estimate along with the value of the result. Therefore, structures are provided for amalgamating a value and error estimate. These structures are declared in the header file gsl_sf_result.h.
The gsl_sf_result struct contains value and error fields.
typedef struct
{
double val;
double err;
} gsl_sf_result;
The field val contains the value and the field err contains an estimate of the absolute error in the value.
In some cases, an overflow or underflow can be detected and handled by a
function. In this case, it may be possible to return a scaling exponent
as well as an error/value pair in order to save the result from
exceeding the dynamic range of the built-in types. The
gsl_sf_result_e10 struct contains value and error fields as well
as an exponent field such that the actual result is obtained as
result * 10^(e10).
typedef struct
{
double val;
double err;
int e10;
} gsl_sf_result_e10;
Previous: Special Functions Examples, Up: Special Functions [Index]
The library follows the conventions of Abramowitz & Stegun where possible,
The following papers contain information on the algorithms used to compute the special functions,
Any errors reported by the library are passed to the function
gsl_error. By running your programs under gdb and setting a
breakpoint in this function you can automatically catch any library
errors. You can add a breakpoint for every session by putting
break gsl_error
into your .gdbinit file in the directory where your program is started.
If the breakpoint catches an error then you can use a backtrace
(bt) to see the call-tree, and the arguments which possibly
caused the error. By moving up into the calling function you can
investigate the values of variables at that point. Here is an example
from the program fft/test_trap, which contains the following
line,
status = gsl_fft_complex_wavetable_alloc (0, &complex_wavetable);
The function gsl_fft_complex_wavetable_alloc takes the length of
an FFT as its first argument. When this line is executed an error will
be generated because the length of an FFT is not allowed to be zero.
To debug this problem we start gdb, using the file
.gdbinit to define a breakpoint in gsl_error,
$ gdb test_trap GDB is free software and you are welcome to distribute copies of it under certain conditions; type "show copying" to see the conditions. There is absolutely no warranty for GDB; type "show warranty" for details. GDB 4.16 (i586-debian-linux), Copyright 1996 Free Software Foundation, Inc. Breakpoint 1 at 0x8050b1e: file error.c, line 14.
When we run the program this breakpoint catches the error and shows the reason for it.
(gdb) run
Starting program: test_trap
Breakpoint 1, gsl_error (reason=0x8052b0d
"length n must be positive integer",
file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
at error.c:14
14 if (gsl_error_handler)
The first argument of gsl_error is always a string describing the
error. Now we can look at the backtrace to see what caused the problem,
(gdb) bt
#0 gsl_error (reason=0x8052b0d
"length n must be positive integer",
file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
at error.c:14
#1 0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
wavetable=0xbffff778) at c_init.c:108
#2 0x8048a00 in main (argc=1, argv=0xbffff9bc)
at test_trap.c:94
#3 0x80488be in ___crt_dummy__ ()
We can see that the error was generated in the function
gsl_fft_complex_wavetable_alloc when it was called with an
argument of n=0. The original call came from line 94 in the
file test_trap.c.
By moving up to the level of the original call we can find the line that caused the error,
(gdb) up
#1 0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
wavetable=0xbffff778) at c_init.c:108
108 GSL_ERROR ("length n must be positive integer", GSL_EDOM);
(gdb) up
#2 0x8048a00 in main (argc=1, argv=0xbffff9bc)
at test_trap.c:94
94 status = gsl_fft_complex_wavetable_alloc (0,
&complex_wavetable);
Thus we have found the line that caused the problem. From this point we
could also print out the values of other variables such as
complex_wavetable.
Previous: Discrete Hankel Transform Functions, Up: Discrete Hankel Transforms [Index]
The algorithms used by these functions are described in the following papers,
Next: Exchanging elements, Previous: Vector views, Up: Vectors [Index]
Common operations on vectors such as addition and multiplication are available in the BLAS part of the library (see BLAS Support). However, it is useful to have a small number of utility functions which do not require the full BLAS code. The following functions fall into this category.
This function copies the elements of the vector src into the vector dest. The two vectors must have the same length.
This function exchanges the elements of the vectors v and w by copying. The two vectors must have the same length.
Next: 6-j Symbols, Up: Coupling Coefficients [Index]
These routines compute the Wigner 3-j coefficient,
(ja jb jc ma mb mc)
where the arguments are given in half-integer units, ja = two_ja/2, ma = two_ma/2, etc.
Next: Quadratic Equations, Previous: Polynomial Evaluation, Up: Polynomials [Index]
The functions described here manipulate polynomials stored in Newton’s divided-difference representation. The use of divided-differences is described in Abramowitz & Stegun sections 25.1.4 and 25.2.26, and Burden and Faires, chapter 3, and discussed briefly below.
Given a function f(x), an nth degree interpolating polynomial P_{n}(x) can be constructed which agrees with f at n+1 distinct points x_0,x_1,...,x_{n}. This polynomial can be written in a form known as Newton’s divided-difference representation:
P_n(x) = f(x_0) + \sum_(k=1)^n [x_0,x_1,...,x_k] (x-x_0)(x-x_1)...(x-x_(k-1))
where the divided differences [x_0,x_1,...,x_k] are defined in section 25.1.4 of Abramowitz and Stegun. Additionally, it is possible to construct an interpolating polynomial of degree 2n+1 which also matches the first derivatives of f at the points x_0,x_1,...,x_n. This is called the Hermite interpolating polynomial and is defined as
H_(2n+1)(x) = f(z_0) + \sum_(k=1)^(2n+1) [z_0,z_1,...,z_k] (x-z_0)(x-z_1)...(x-z_(k-1))
where the elements of z = \{x_0,x_0,x_1,x_1,...,x_n,x_n\} are defined by z_{2k} = z_{2k+1} = x_k. The divided-differences [z_0,z_1,...,z_k] are discussed in Burden and Faires, section 3.4.
This function computes a divided-difference representation of the interpolating polynomial for the points (x, y) stored in the arrays xa and ya of length size. On output the divided-differences of (xa,ya) are stored in the array dd, also of length size. Using the notation above, dd[k] = [x_0,x_1,...,x_k].
This function evaluates the polynomial stored in divided-difference form
in the arrays dd and xa of length size at the point
x. An inline version of this function is used when HAVE_INLINE is defined.
This function converts the divided-difference representation of a polynomial to a Taylor expansion. The divided-difference representation is supplied in the arrays dd and xa of length size. On output the Taylor coefficients of the polynomial expanded about the point xp are stored in the array c also of length size. A workspace of length size must be provided in the array w.
This function computes a divided-difference representation of the
interpolating Hermite polynomial for the points (x, y) stored in
the arrays xa and ya of length size. Hermite interpolation
constructs polynomials which also match first derivatives dy/dx which are
provided in the array dya also of length size. The first derivatives can be
incorported into the usual divided-difference algorithm by forming a new
dataset z = \{x_0,x_0,x_1,x_1,...\}, which is stored in the array
za of length 2*size on output. On output the
divided-differences of the Hermite representation are stored in the array
dd, also of length 2*size. Using the notation above,
dd[k] = [z_0,z_1,...,z_k]. The resulting Hermite polynomial
can be evaluated by calling gsl_poly_dd_eval and using
za for the input argument xa.
Next: Quadratic Equations, Previous: Polynomial Evaluation, Up: Polynomials [Index]
Next: DWT Transform Functions, Previous: DWT Definitions, Up: Wavelet Transforms [Index]
The gsl_wavelet structure contains the filter coefficients
defining the wavelet and any associated offset parameters.
This function allocates and initializes a wavelet object of type T. The parameter k selects the specific member of the wavelet family. A null pointer is returned if insufficient memory is available or if a unsupported member is selected.
The following wavelet types are implemented:
This is the Daubechies wavelet family of maximum phase with k/2 vanishing moments. The implemented wavelets are k=4, 6, …, 20, with k even.
This is the Haar wavelet. The only valid choice of k for the Haar wavelet is k=2.
This is the biorthogonal B-spline wavelet family of order (i,j). The implemented values of k = 100*i + j are 103, 105, 202, 204, 206, 208, 301, 303, 305 307, 309.
The centered forms of the wavelets align the coefficients of the various sub-bands on edges. Thus the resulting visualization of the coefficients of the wavelet transform in the phase plane is easier to understand.
This function returns a pointer to the name of the wavelet family for w.
The gsl_wavelet_workspace structure contains scratch space of the
same size as the input data and is used to hold intermediate results
during the transform.
This function allocates a workspace for the discrete wavelet transform. To perform a one-dimensional transform on n elements, a workspace of size n must be provided. For two-dimensional transforms of n-by-n matrices it is sufficient to allocate a workspace of size n, since the transform operates on individual rows and columns. A null pointer is returned if insufficient memory is available.
This function frees the allocated workspace work.
Next: DWT Transform Functions, Previous: DWT Definitions, Up: Wavelet Transforms [Index]
Next: Random number generator algorithms, Previous: Copying random number generator state, Up: Random Number Generation [Index]
The library provides functions for reading and writing the random number state to a file as binary data.
This function writes the random number state of the random number
generator r to the stream stream in binary format. The
return value is 0 for success and GSL_EFAILED if there was a
problem writing to the file. Since the data is written in the native
binary format it may not be portable between different architectures.
This function reads the random number state into the random number
generator r from the open stream stream in binary format.
The random number generator r must be preinitialized with the
correct random number generator type since type information is not
saved. The return value is 0 for success and GSL_EFAILED if
there was a problem reading from the file. The data is assumed to
have been written in the native binary format on the same
architecture.
Next: The Laplace Distribution, Previous: The Multivariate Gaussian Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the exponential distribution with mean mu. The distribution is,
p(x) dx = {1 \over \mu} \exp(-x/\mu) dx
for x >= 0.
This function computes the probability density p(x) at x for an exponential distribution with mean mu, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the exponential distribution with mean mu.
The algorithms for general functions (without a weight function) are based on Gauss-Kronrod rules.
A Gauss-Kronrod rule begins with a classical Gaussian quadrature rule of order m. This is extended with additional points between each of the abscissae to give a higher order Kronrod rule of order 2m+1. The Kronrod rule is efficient because it reuses existing function evaluations from the Gaussian rule.
The higher order Kronrod rule is used as the best approximation to the integral, and the difference between the two rules is used as an estimate of the error in the approximation.
gsl-ref-html-2.3/Creating-ntuples.html 0000664 0001750 0001750 00000010217 13055414474 016055 0 ustar edd eddNext: Opening an existing ntuple file, Previous: The ntuple struct, Up: N-tuples [Index]
This function creates a new write-only ntuple file filename for ntuples of size size and returns a pointer to the newly created ntuple struct. Any existing file with the same name is truncated to zero length and overwritten. A pointer to memory for the current ntuple row ntuple_data must be supplied—this is used to copy ntuples in and out of the file.
Next: The F-distribution, Previous: The Lognormal Distribution, Up: Random Number Distributions [Index]
The chi-squared distribution arises in statistics. If Y_i are n independent Gaussian random variates with unit variance then the sum-of-squares,
X_i = \sum_i Y_i^2
has a chi-squared distribution with n degrees of freedom.
This function returns a random variate from the chi-squared distribution with nu degrees of freedom. The distribution function is,
p(x) dx = {1 \over 2 \Gamma(\nu/2) } (x/2)^{\nu/2 - 1} \exp(-x/2) dx
for x >= 0.
This function computes the probability density p(x) at x for a chi-squared distribution with nu degrees of freedom, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the chi-squared distribution with nu degrees of freedom.
Next: Root Finding Caveats, Up: One dimensional Root-Finding [Index]
One-dimensional root finding algorithms can be divided into two classes, root bracketing and root polishing. Algorithms which proceed by bracketing a root are guaranteed to converge. Bracketing algorithms begin with a bounded region known to contain a root. The size of this bounded region is reduced, iteratively, until it encloses the root to a desired tolerance. This provides a rigorous error estimate for the location of the root.
The technique of root polishing attempts to improve an initial guess to the root. These algorithms converge only if started “close enough” to a root, and sacrifice a rigorous error bound for speed. By approximating the behavior of a function in the vicinity of a root they attempt to find a higher order improvement of an initial guess. When the behavior of the function is compatible with the algorithm and a good initial guess is available a polishing algorithm can provide rapid convergence.
In GSL both types of algorithm are available in similar frameworks. The user provides a high-level driver for the algorithms, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,
The state for bracketing solvers is held in a gsl_root_fsolver
struct. The updating procedure uses only function evaluations (not
derivatives). The state for root polishing solvers is held in a
gsl_root_fdfsolver struct. The updates require both the function
and its derivative (hence the name fdf) to be supplied by the
user.
Next: Root Finding Caveats, Up: One dimensional Root-Finding [Index]
Next: Saving and restoring quasi-random number generator state, Previous: Sampling from a quasi-random number generator, Up: Quasi-Random Sequences [Index]
This function returns a pointer to the name of the generator.
These functions return a pointer to the state of generator r and its size. You can use this information to access the state directly. For example, the following code will write the state of a generator to a stream,
void * state = gsl_qrng_state (q); size_t n = gsl_qrng_size (q); fwrite (state, n, 1, stream);
Previous: Sparse Matrices Examples, Up: Sparse Matrices [Index]
The algorithms used by these functions are described in the following sources:
Next: Conical Functions, Previous: Legendre Polynomials, Up: Legendre Functions and Spherical Harmonics [Index]
The following functions compute the associated Legendre polynomials P_l^m(x) which are solutions of the differential equation
(1 - x^2) d^2 P_l^m(x) / dx^2 P_l^m(x) - 2x d/dx P_l^m(x) + ( l(l+1) - m^2 / (1 - x^2) ) P_l^m(x) = 0
where the degree l and order m satisfy 0 \le l and 0 \le m \le l. The functions P_l^m(x) grow combinatorially with l and can overflow for l larger than about 150. Alternatively, one may calculate normalized associated Legendre polynomials. There are a number of different normalization conventions, and these functions can be stably computed up to degree and order 2700. The following normalizations are provided:
Schmidt semi-normalizationSchmidt semi-normalized associated Legendre polynomials are often used in the magnetics community and are defined as
S_l^0(x) = P_l^0(x) S_l^m(x) = (-1)^m \sqrt((2(l-m)! / (l+m)!)) P_l^m(x), m > 0
The factor of (-1)^m is called the Condon-Shortley phase
factor and can be excluded if desired by setting the parameter
csphase = 1 in the functions below.
Spherical Harmonic NormalizationThe associated Legendre polynomials suitable for calculating spherical harmonics are defined as
Y_l^m(x) = (-1)^m \sqrt((2l + 1) * (l-m)! / (4 \pi) / (l+m)!) P_l^m(x)
where again the phase factor (-1)^m can be included or excluded if desired.
Full NormalizationThe fully normalized associated Legendre polynomials are defined as
N_l^m(x) = (-1)^m \sqrt((l + 1/2) * (l-m)! / (l+m)!) P_l^m(x)
and have the property
\int_(-1)^1 ( N_l^m(x) )^2 dx = 1
The normalized associated Legendre routines below use a recurrence relation which is stable up to a degree and order of about 2700. Beyond this, the computed functions could suffer from underflow leading to incorrect results. Routines are provided to compute first and second derivatives dP_l^m(x)/dx and d^2 P_l^m(x)/dx^2 as well as their alternate versions d P_l^m(\cos{\theta})/d\theta and d^2 P_l^m(\cos{\theta})/d\theta^2. While there is a simple scaling relationship between the two forms, the derivatives involving \theta are heavily used in spherical harmonic expansions and so these routines are also provided.
In the functions below, a parameter of type gsl_sf_legendre_t
specifies the type of normalization to use. The possible values are
GSL_SF_LEGENDRE_NONEThis specifies the computation of the unnormalized associated Legendre polynomials P_l^m(x).
GSL_SF_LEGENDRE_SCHMIDTThis specifies the computation of the Schmidt semi-normalized associated Legendre polynomials S_l^m(x).
GSL_SF_LEGENDRE_SPHARMThis specifies the computation of the spherical harmonic associated Legendre polynomials Y_l^m(x).
GSL_SF_LEGENDRE_FULLThis specifies the computation of the fully normalized associated Legendre polynomials N_l^m(x).
These functions calculate all normalized associated Legendre
polynomials for 0 \le l \le lmax and
0 \le m \le l for
|x| <= 1.
The norm parameter specifies which normalization is used.
The normalized P_l^m(x) values are stored in result_array, whose
minimum size can be obtained from calling gsl_sf_legendre_array_n.
The array index of P_l^m(x) is obtained from calling
gsl_sf_legendre_array_index(l, m). To include or exclude
the Condon-Shortley phase factor of (-1)^m, set the parameter
csphase to either -1 or 1 respectively in the
_e function. This factor is included by default.
These functions calculate all normalized associated Legendre
functions and their first derivatives up to degree lmax for
|x| < 1.
The parameter norm specifies the normalization used. The
normalized P_l^m(x) values and their derivatives
dP_l^m(x)/dx are stored in result_array and
result_deriv_array respectively.
To include or exclude
the Condon-Shortley phase factor of (-1)^m, set the parameter
csphase to either -1 or 1 respectively in the
_e function. This factor is included by default.
These functions calculate all normalized associated Legendre
functions and their (alternate) first derivatives up to degree lmax for
|x| < 1.
The normalized P_l^m(x) values and their derivatives
dP_l^m(\cos{\theta})/d\theta are stored in result_array and
result_deriv_array respectively.
To include or exclude
the Condon-Shortley phase factor of (-1)^m, set the parameter
csphase to either -1 or 1 respectively in the
_e function. This factor is included by default.
These functions calculate all normalized associated Legendre
functions and their first and second derivatives up to degree lmax for
|x| < 1.
The parameter norm specifies the normalization used. The
normalized P_l^m(x), their first derivatives
dP_l^m(x)/dx, and their second derivatives
d^2 P_l^m(x)/dx^2 are stored in result_array,
result_deriv_array, and result_deriv2_array respectively.
To include or exclude
the Condon-Shortley phase factor of (-1)^m, set the parameter
csphase to either -1 or 1 respectively in the
_e function. This factor is included by default.
These functions calculate all normalized associated Legendre
functions and their (alternate) first and second derivatives up to degree
lmax for
|x| < 1.
The parameter norm specifies the normalization used. The
normalized P_l^m(x), their first derivatives
dP_l^m(\cos{\theta})/d\theta, and their second derivatives
d^2 P_l^m(\cos{\theta})/d\theta^2 are stored in result_array,
result_deriv_array, and result_deriv2_array respectively.
To include or exclude
the Condon-Shortley phase factor of (-1)^m, set the parameter
csphase to either -1 or 1 respectively in the
_e function. This factor is included by default.
This function returns the minimum array size for maximum degree lmax needed for the array versions of the associated Legendre functions. Size is calculated as the total number of P_l^m(x) functions, plus extra space for precomputing multiplicative factors used in the recurrence relations.
This function returns the index into result_array, result_deriv_array, or result_deriv2_array corresponding to P_l^m(x), P_l^{'m}(x), or P_l^{''m}(x). The index is given by l(l+1)/2 + m.
These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |x| <= 1.
These routines compute the normalized associated Legendre polynomial \sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x) suitable for use in spherical harmonics. The parameters must satisfy m >= 0, l >= m, |x| <= 1. Theses routines avoid the overflows that occur for the standard normalization of P_l^m(x).
These functions are now deprecated and will be removed in a future
release; see gsl_sf_legendre_array and
gsl_sf_legendre_deriv_array.
These functions are now deprecated and will be removed in a future
release; see gsl_sf_legendre_array and
gsl_sf_legendre_deriv_array.
This function is now deprecated and will be removed in a future release.
Next: Conical Functions, Previous: Legendre Polynomials, Up: Legendre Functions and Spherical Harmonics [Index]
Next: Special Functions Examples, Previous: Trigonometric Functions, Up: Special Functions [Index]
The Riemann zeta function is defined in Abramowitz & Stegun, Section 23.2. The functions described in this section are declared in the header file gsl_sf_zeta.h.
| • Riemann Zeta Function: | ||
| • Riemann Zeta Function Minus One: | ||
| • Hurwitz Zeta Function: | ||
| • Eta Function: |
Next: Histogram allocation, Up: Histograms [Index]
A histogram is defined by the following struct,
size_t nThis is the number of histogram bins
double * rangeThe ranges of the bins are stored in an array of n+1 elements pointed to by range.
double * binThe counts for each bin are stored in an array of n elements pointed to by bin. The bins are floating-point numbers, so you can increment them by non-integer values if necessary.
The range for bin[i] is given by range[i] to range[i+1]. For n bins there are n+1 entries in the array range. Each bin is inclusive at the lower end and exclusive at the upper end. Mathematically this means that the bins are defined by the following inequality,
bin[i] corresponds to range[i] <= x < range[i+1]
Here is a diagram of the correspondence between ranges and bins on the number-line for x,
[ bin[0] )[ bin[1] )[ bin[2] )[ bin[3] )[ bin[4] )
---|---------|---------|---------|---------|---------|--- x
r[0] r[1] r[2] r[3] r[4] r[5]
In this picture the values of the range array are denoted by r. On the left-hand side of each bin the square bracket ‘[’ denotes an inclusive lower bound (r <= x), and the round parentheses ‘)’ on the right-hand side denote an exclusive upper bound (x < r). Thus any samples which fall on the upper end of the histogram are excluded. If you want to include this value for the last bin you will need to add an extra bin to your histogram.
The gsl_histogram struct and its associated functions are defined
in the header file gsl_histogram.h.
Next: Histogram allocation, Up: Histograms [Index]
Next: ODE References and Further Reading, Previous: Driver, Up: Ordinary Differential Equations [Index]
The following program solves the second-order nonlinear Van der Pol oscillator equation,
u''(t) + \mu u'(t) (u(t)^2 - 1) + u(t) = 0
This can be converted into a first order system suitable for use with the routines described in this chapter by introducing a separate variable for the velocity, v = u'(t),
u' = v v' = -u + \mu v (1-u^2)
The program begins by defining functions for these derivatives and their Jacobian. The main function uses driver level functions to solve the problem. The program evolves the solution from (u, v) = (1, 0) at t=0 to t=100. The step-size h is automatically adjusted by the controller to maintain an absolute accuracy of 10^{-6} in the function values (u, v). The loop in the example prints the solution at the points t_i = 1, 2, \dots, 100.
#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_odeiv2.h>
int
func (double t, const double y[], double f[],
void *params)
{
(void)(t); /* avoid unused parameter warning */
double mu = *(double *)params;
f[0] = y[1];
f[1] = -y[0] - mu*y[1]*(y[0]*y[0] - 1);
return GSL_SUCCESS;
}
int
jac (double t, const double y[], double *dfdy,
double dfdt[], void *params)
{
(void)(t); /* avoid unused parameter warning */
double mu = *(double *)params;
gsl_matrix_view dfdy_mat
= gsl_matrix_view_array (dfdy, 2, 2);
gsl_matrix * m = &dfdy_mat.matrix;
gsl_matrix_set (m, 0, 0, 0.0);
gsl_matrix_set (m, 0, 1, 1.0);
gsl_matrix_set (m, 1, 0, -2.0*mu*y[0]*y[1] - 1.0);
gsl_matrix_set (m, 1, 1, -mu*(y[0]*y[0] - 1.0));
dfdt[0] = 0.0;
dfdt[1] = 0.0;
return GSL_SUCCESS;
}
int
main (void)
{
double mu = 10;
gsl_odeiv2_system sys = {func, jac, 2, &mu};
gsl_odeiv2_driver * d =
gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk8pd,
1e-6, 1e-6, 0.0);
int i;
double t = 0.0, t1 = 100.0;
double y[2] = { 1.0, 0.0 };
for (i = 1; i <= 100; i++)
{
double ti = i * t1 / 100.0;
int status = gsl_odeiv2_driver_apply (d, &t, ti, y);
if (status != GSL_SUCCESS)
{
printf ("error, return value=%d\n", status);
break;
}
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}
gsl_odeiv2_driver_free (d);
return 0;
}
The user can work with the lower level functions directly, as in the following example. In this case an intermediate result is printed after each successful step instead of equidistant time points.
int
main (void)
{
const gsl_odeiv2_step_type * T
= gsl_odeiv2_step_rk8pd;
gsl_odeiv2_step * s
= gsl_odeiv2_step_alloc (T, 2);
gsl_odeiv2_control * c
= gsl_odeiv2_control_y_new (1e-6, 0.0);
gsl_odeiv2_evolve * e
= gsl_odeiv2_evolve_alloc (2);
double mu = 10;
gsl_odeiv2_system sys = {func, jac, 2, &mu};
double t = 0.0, t1 = 100.0;
double h = 1e-6;
double y[2] = { 1.0, 0.0 };
while (t < t1)
{
int status = gsl_odeiv2_evolve_apply (e, c, s,
&sys,
&t, t1,
&h, y);
if (status != GSL_SUCCESS)
break;
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}
gsl_odeiv2_evolve_free (e);
gsl_odeiv2_control_free (c);
gsl_odeiv2_step_free (s);
return 0;
}
For functions with multiple parameters, the appropriate information
can be passed in through the params argument in
gsl_odeiv2_system definition (mu in this example) by using
a pointer to a struct.
It is also possible to work with a non-adaptive integrator, using only
the stepping function itself,
gsl_odeiv2_driver_apply_fixed_step or
gsl_odeiv2_evolve_apply_fixed_step. The following program uses
the driver level function, with fourth-order
Runge-Kutta stepping function with a fixed stepsize of
0.001.
int
main (void)
{
double mu = 10;
gsl_odeiv2_system sys = { func, jac, 2, &mu };
gsl_odeiv2_driver *d =
gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk4,
1e-3, 1e-8, 1e-8);
double t = 0.0;
double y[2] = { 1.0, 0.0 };
int i, s;
for (i = 0; i < 100; i++)
{
s = gsl_odeiv2_driver_apply_fixed_step (d, &t, 1e-3, 1000, y);
if (s != GSL_SUCCESS)
{
printf ("error: driver returned %d\n", s);
break;
}
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}
gsl_odeiv2_driver_free (d);
return s;
}
Next: ODE References and Further Reading, Previous: Driver, Up: Ordinary Differential Equations [Index]
Next: Running Statistics Example programs, Previous: Running Statistics Current Statistics, Up: Running Statistics [Index]
The functions in this section estimate quantiles dynamically without storing the entire dataset, using the algorithm of Jain and Chlamtec, 1985. Only five points (markers) are stored which represent the minimum and maximum of the data, as well as current estimates of the p/2-, p-, and (1+p)/2-quantiles. Each time a new data point is added, the marker positions and heights are updated.
This function allocates a workspace for the dynamic estimation of p-quantiles, where p is between 0 and 1. The median corresponds to p = 0.5. The size of the workspace is O(1).
This function frees the memory associated with the workspace w.
This function resets the workspace w to its initial state, so it can begin working on a new set of data.
This function updates the estimate of the p-quantile with the new data point x.
This function returns the current estimate of the p-quantile.
Next: Vector views, Previous: Initializing vector elements, Up: Vectors [Index]
The library provides functions for reading and writing vectors to a file as binary data or formatted text.
This function writes the elements of the vector v to the stream
stream in binary format. The return value is 0 for success and
GSL_EFAILED if there was a problem writing to the file. Since the
data is written in the native binary format it may not be portable
between different architectures.
This function reads into the vector v from the open stream
stream in binary format. The vector v must be preallocated
with the correct length since the function uses the size of v to
determine how many bytes to read. The return value is 0 for success and
GSL_EFAILED if there was a problem reading from the file. The
data is assumed to have been written in the native binary format on the
same architecture.
This function writes the elements of the vector v line-by-line to
the stream stream using the format specifier format, which
should be one of the %g, %e or %f formats for
floating point numbers and %d for integers. The function returns
0 for success and GSL_EFAILED if there was a problem writing to
the file.
This function reads formatted data from the stream stream into the
vector v. The vector v must be preallocated with the correct
length since the function uses the size of v to determine how many
numbers to read. The function returns 0 for success and
GSL_EFAILED if there was a problem reading from the file.
Next: Vector views, Previous: Initializing vector elements, Up: Vectors [Index]
Next: Elementary Complex Functions, Previous: Properties of complex numbers, Up: Complex Numbers [Index]
This function returns the sum of the complex numbers a and b, z=a+b.
This function returns the difference of the complex numbers a and b, z=a-b.
This function returns the product of the complex numbers a and b, z=ab.
This function returns the quotient of the complex numbers a and b, z=a/b.
This function returns the sum of the complex number a and the real number x, z=a+x.
This function returns the difference of the complex number a and the real number x, z=a-x.
This function returns the product of the complex number a and the real number x, z=ax.
This function returns the quotient of the complex number a and the real number x, z=a/x.
This function returns the sum of the complex number a and the imaginary number iy, z=a+iy.
This function returns the difference of the complex number a and the imaginary number iy, z=a-iy.
This function returns the product of the complex number a and the imaginary number iy, z=a*(iy).
This function returns the quotient of the complex number a and the imaginary number iy, z=a/(iy).
This function returns the complex conjugate of the complex number z, z^* = x - i y.
This function returns the inverse, or reciprocal, of the complex number z, 1/z = (x - i y)/(x^2 + y^2).
This function returns the negative of the complex number z, -z = (-x) + i(-y).
Next: Elementary Complex Functions, Previous: Properties of complex numbers, Up: Complex Numbers [Index]
Next: Ei_3(x), Previous: Ei(x), Up: Exponential Integrals [Index]
These routines compute the integral Shi(x) = \int_0^x dt \sinh(t)/t.
These routines compute the integral Chi(x) := \Re[ \gamma_E + \log(x) + \int_0^x dt (\cosh(t)-1)/t] ,
where \gamma_E is the Euler constant (available as the macro M_EULER).
Next: Debugging References, Previous: Handling floating point exceptions, Up: Debugging Numerical Programs [Index]
Writing reliable numerical programs in C requires great care. The following GCC warning options are recommended when compiling numerical programs:
gcc -ansi -pedantic -Werror -Wall -W -Wmissing-prototypes -Wstrict-prototypes -Wconversion -Wshadow -Wpointer-arith -Wcast-qual -Wcast-align -Wwrite-strings -Wnested-externs -fshort-enums -fno-common -Dinline= -g -O2
For details of each option consult the manual Using and Porting GCC. The following table gives a brief explanation of what types of errors these options catch.
-ansi -pedanticUse ANSI C, and reject any non-ANSI extensions. These flags help in writing portable programs that will compile on other systems.
-WerrorConsider warnings to be errors, so that compilation stops. This prevents warnings from scrolling off the top of the screen and being lost. You won’t be able to compile the program until it is completely warning-free.
-WallThis turns on a set of warnings for common programming problems. You
need -Wall, but it is not enough on its own.
-O2Turn on optimization. The warnings for uninitialized variables in
-Wall rely on the optimizer to analyze the code. If there is no
optimization then these warnings aren’t generated.
-WThis turns on some extra warnings not included in -Wall, such as
missing return values and comparisons between signed and unsigned
integers.
-Wmissing-prototypes -Wstrict-prototypesWarn if there are any missing or inconsistent prototypes. Without prototypes it is harder to detect problems with incorrect arguments.
-WconversionThe main use of this option is to warn about conversions from signed to
unsigned integers. For example, unsigned int x = -1. If you need
to perform such a conversion you can use an explicit cast.
-WshadowThis warns whenever a local variable shadows another local variable. If two variables have the same name then it is a potential source of confusion.
-Wpointer-arith -Wcast-qual -Wcast-alignThese options warn if you try to do pointer arithmetic for types which
don’t have a size, such as void, if you remove a const
cast from a pointer, or if you cast a pointer to a type which has a
different size, causing an invalid alignment.
-Wwrite-stringsThis option gives string constants a const qualifier so that it
will be a compile-time error to attempt to overwrite them.
-fshort-enumsThis option makes the type of enum as short as possible. Normally
this makes an enum different from an int. Consequently any
attempts to assign a pointer-to-int to a pointer-to-enum will generate a
cast-alignment warning.
-fno-commonThis option prevents global variables being simultaneously defined in
different object files (you get an error at link time). Such a variable
should be defined in one file and referred to in other files with an
extern declaration.
-Wnested-externsThis warns if an extern declaration is encountered within a
function.
-Dinline=The inline keyword is not part of ANSI C. Thus if you want to use
-ansi with a program which uses inline functions you can use this
preprocessor definition to remove the inline keywords.
-gIt always makes sense to put debugging symbols in the executable so that
you can debug it using gdb. The only effect of debugging symbols
is to increase the size of the file, and you can use the strip
command to remove them later if necessary.
Next: Debugging References, Previous: Handling floating point exceptions, Up: Debugging Numerical Programs [Index]
Next: Radial Functions for Hyperbolic Space, Previous: Associated Legendre Polynomials and Spherical Harmonics, Up: Legendre Functions and Spherical Harmonics [Index]
The Conical Functions P^\mu_{-(1/2)+i\lambda}(x) and Q^\mu_{-(1/2)+i\lambda} are described in Abramowitz & Stegun, Section 8.12.
These routines compute the irregular Spherical Conical Function P^{1/2}_{-1/2 + i \lambda}(x) for x > -1.
These routines compute the regular Spherical Conical Function P^{-1/2}_{-1/2 + i \lambda}(x) for x > -1.
These routines compute the conical function P^0_{-1/2 + i \lambda}(x) for x > -1.
These routines compute the conical function P^1_{-1/2 + i \lambda}(x) for x > -1.
These routines compute the Regular Spherical Conical Function P^{-1/2-l}_{-1/2 + i \lambda}(x) for x > -1, l >= -1.
These routines compute the Regular Cylindrical Conical Function P^{-m}_{-1/2 + i \lambda}(x) for x > -1, m >= -1.
The IEEE Standard for Binary Floating-Point Arithmetic defines binary formats for single and double precision numbers. Each number is composed of three parts: a sign bit (s), an exponent (E) and a fraction (f). The numerical value of the combination (s,E,f) is given by the following formula,
(-1)^s (1.fffff...) 2^E
The sign bit is either zero or one. The exponent ranges from a minimum value E_min to a maximum value E_max depending on the precision. The exponent is converted to an unsigned number e, known as the biased exponent, for storage by adding a bias parameter, e = E + bias. The sequence fffff... represents the digits of the binary fraction f. The binary digits are stored in normalized form, by adjusting the exponent to give a leading digit of 1. Since the leading digit is always 1 for normalized numbers it is assumed implicitly and does not have to be stored. Numbers smaller than 2^(E_min) are be stored in denormalized form with a leading zero,
(-1)^s (0.fffff...) 2^(E_min)
This allows gradual underflow down to 2^(E_min - p) for p bits of precision. A zero is encoded with the special exponent of 2^(E_min - 1) and infinities with the exponent of 2^(E_max + 1).
The format for single precision numbers uses 32 bits divided in the following way,
seeeeeeeefffffffffffffffffffffff
s = sign bit, 1 bit
e = exponent, 8 bits (E_min=-126, E_max=127, bias=127)
f = fraction, 23 bits
The format for double precision numbers uses 64 bits divided in the following way,
seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffff s = sign bit, 1 bit e = exponent, 11 bits (E_min=-1022, E_max=1023, bias=1023) f = fraction, 52 bits
It is often useful to be able to investigate the behavior of a calculation at the bit-level and the library provides functions for printing the IEEE representations in a human-readable form.
These functions output a formatted version of the IEEE floating-point
number pointed to by x to the stream stream. A pointer is
used to pass the number indirectly, to avoid any undesired promotion
from float to double. The output takes one of the
following forms,
NaNthe Not-a-Number symbol
Inf, -Infpositive or negative infinity
1.fffff...*2^E, -1.fffff...*2^Ea normalized floating point number
0.fffff...*2^E, -0.fffff...*2^Ea denormalized floating point number
0, -0positive or negative zero
The output can be used directly in GNU Emacs Calc mode by preceding it
with 2# to indicate binary.
These functions output a formatted version of the IEEE floating-point
number pointed to by x to the stream stdout.
The following program demonstrates the use of the functions by printing the single and double precision representations of the fraction 1/3. For comparison the representation of the value promoted from single to double precision is also printed.
#include <stdio.h>
#include <gsl/gsl_ieee_utils.h>
int
main (void)
{
float f = 1.0/3.0;
double d = 1.0/3.0;
double fd = f; /* promote from float to double */
printf (" f="); gsl_ieee_printf_float(&f);
printf ("\n");
printf ("fd="); gsl_ieee_printf_double(&fd);
printf ("\n");
printf (" d="); gsl_ieee_printf_double(&d);
printf ("\n");
return 0;
}
The binary representation of 1/3 is 0.01010101... . The output below shows that the IEEE format normalizes this fraction to give a leading digit of 1,
f= 1.01010101010101010101011*2^-2 fd= 1.0101010101010101010101100000000000000000000000000000*2^-2 d= 1.0101010101010101010101010101010101010101010101010101*2^-2
The output also shows that a single-precision number is promoted to double-precision by adding zeros in the binary representation.
This function computes the matrix-vector product and sum
y \leftarrow \alpha op(A) x + \beta y, where
op(A) = A, A^T for TransA = CblasNoTrans,
CblasTrans. In-place computations are not supported, so
x and y must be distinct vectors.
The matrix A may be in triplet or compressed format.
This function computes the sparse matrix-matrix product C = \alpha A B. The matrices must be in compressed format.
Next: Permutations in cyclic form, Previous: Applying Permutations, Up: Permutations [Index]
The library provides functions for reading and writing permutations to a file as binary data or formatted text.
This function writes the elements of the permutation p to the
stream stream in binary format. The function returns
GSL_EFAILED if there was a problem writing to the file. Since the
data is written in the native binary format it may not be portable
between different architectures.
This function reads into the permutation p from the open stream
stream in binary format. The permutation p must be
preallocated with the correct length since the function uses the size of
p to determine how many bytes to read. The function returns
GSL_EFAILED if there was a problem reading from the file. The
data is assumed to have been written in the native binary format on the
same architecture.
This function writes the elements of the permutation p
line-by-line to the stream stream using the format specifier
format, which should be suitable for a type of size_t.
In ISO C99 the type modifier z represents size_t, so
"%zu\n" is a suitable format.9
The function returns GSL_EFAILED if there was a problem writing
to the file.
This function reads formatted data from the stream stream into the
permutation p. The permutation p must be preallocated with
the correct length since the function uses the size of p to
determine how many numbers to read. The function returns
GSL_EFAILED if there was a problem reading from the file.
In versions of the
GNU C library prior to the ISO C99 standard,
the type modifier Z was used instead.
Next: Permutations in cyclic form, Previous: Applying Permutations, Up: Permutations [Index]
Next: Monte Carlo Integration References and Further Reading, Previous: VEGAS, Up: Monte Carlo Integration [Index]
The example program below uses the Monte Carlo routines to estimate the value of the following 3-dimensional integral from the theory of random walks,
I = \int_{-pi}^{+pi} {dk_x/(2 pi)}
\int_{-pi}^{+pi} {dk_y/(2 pi)}
\int_{-pi}^{+pi} {dk_z/(2 pi)}
1 / (1 - cos(k_x)cos(k_y)cos(k_z)).
The analytic value of this integral can be shown to be I = \Gamma(1/4)^4/(4 \pi^3) = 1.393203929685676859.... The integral gives the mean time spent at the origin by a random walk on a body-centered cubic lattice in three dimensions.
For simplicity we will compute the integral over the region (0,0,0) to (\pi,\pi,\pi) and multiply by 8 to obtain the full result. The integral is slowly varying in the middle of the region but has integrable singularities at the corners (0,0,0), (0,\pi,\pi), (\pi,0,\pi) and (\pi,\pi,0). The Monte Carlo routines only select points which are strictly within the integration region and so no special measures are needed to avoid these singularities.
#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_monte.h>
#include <gsl/gsl_monte_plain.h>
#include <gsl/gsl_monte_miser.h>
#include <gsl/gsl_monte_vegas.h>
/* Computation of the integral,
I = int (dx dy dz)/(2pi)^3 1/(1-cos(x)cos(y)cos(z))
over (-pi,-pi,-pi) to (+pi, +pi, +pi). The exact answer
is Gamma(1/4)^4/(4 pi^3). This example is taken from
C.Itzykson, J.M.Drouffe, "Statistical Field Theory -
Volume 1", Section 1.1, p21, which cites the original
paper M.L.Glasser, I.J.Zucker, Proc.Natl.Acad.Sci.USA 74
1800 (1977) */
/* For simplicity we compute the integral over the region
(0,0,0) -> (pi,pi,pi) and multiply by 8 */
double exact = 1.3932039296856768591842462603255;
double
g (double *k, size_t dim, void *params)
{
(void)(dim); /* avoid unused parameter warnings */
(void)(params);
double A = 1.0 / (M_PI * M_PI * M_PI);
return A / (1.0 - cos (k[0]) * cos (k[1]) * cos (k[2]));
}
void
display_results (char *title, double result, double error)
{
printf ("%s ==================\n", title);
printf ("result = % .6f\n", result);
printf ("sigma = % .6f\n", error);
printf ("exact = % .6f\n", exact);
printf ("error = % .6f = %.2g sigma\n", result - exact,
fabs (result - exact) / error);
}
int
main (void)
{
double res, err;
double xl[3] = { 0, 0, 0 };
double xu[3] = { M_PI, M_PI, M_PI };
const gsl_rng_type *T;
gsl_rng *r;
gsl_monte_function G = { &g, 3, 0 };
size_t calls = 500000;
gsl_rng_env_setup ();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
{
gsl_monte_plain_state *s = gsl_monte_plain_alloc (3);
gsl_monte_plain_integrate (&G, xl, xu, 3, calls, r, s,
&res, &err);
gsl_monte_plain_free (s);
display_results ("plain", res, err);
}
{
gsl_monte_miser_state *s = gsl_monte_miser_alloc (3);
gsl_monte_miser_integrate (&G, xl, xu, 3, calls, r, s,
&res, &err);
gsl_monte_miser_free (s);
display_results ("miser", res, err);
}
{
gsl_monte_vegas_state *s = gsl_monte_vegas_alloc (3);
gsl_monte_vegas_integrate (&G, xl, xu, 3, 10000, r, s,
&res, &err);
display_results ("vegas warm-up", res, err);
printf ("converging...\n");
do
{
gsl_monte_vegas_integrate (&G, xl, xu, 3, calls/5, r, s,
&res, &err);
printf ("result = % .6f sigma = % .6f "
"chisq/dof = %.1f\n", res, err, gsl_monte_vegas_chisq (s));
}
while (fabs (gsl_monte_vegas_chisq (s) - 1.0) > 0.5);
display_results ("vegas final", res, err);
gsl_monte_vegas_free (s);
}
gsl_rng_free (r);
return 0;
}
With 500,000 function calls the plain Monte Carlo algorithm achieves a
fractional error of 1%. The estimated error sigma is roughly
consistent with the actual error–the computed result differs from
the true result by about 1.4 standard deviations,
plain ================== result = 1.412209 sigma = 0.013436 exact = 1.393204 error = 0.019005 = 1.4 sigma
The MISER algorithm reduces the error by a factor of four, and also correctly estimates the error,
miser ================== result = 1.391322 sigma = 0.003461 exact = 1.393204 error = -0.001882 = 0.54 sigma
In the case of the VEGAS algorithm the program uses an initial warm-up run of 10,000 function calls to prepare, or “warm up”, the grid. This is followed by a main run with five iterations of 100,000 function calls. The chi-squared per degree of freedom for the five iterations are checked for consistency with 1, and the run is repeated if the results have not converged. In this case the estimates are consistent on the first pass.
vegas warm-up ================== result = 1.392673 sigma = 0.003410 exact = 1.393204 error = -0.000531 = 0.16 sigma converging... result = 1.393281 sigma = 0.000362 chisq/dof = 1.5 vegas final ================== result = 1.393281 sigma = 0.000362 exact = 1.393204 error = 0.000077 = 0.21 sigma
If the value of chisq had differed significantly from 1 it would
indicate inconsistent results, with a correspondingly underestimated
error. The final estimate from VEGAS (using a similar number of
function calls) is significantly more accurate than the other two
algorithms.
Next: Monte Carlo Integration References and Further Reading, Previous: VEGAS, Up: Monte Carlo Integration [Index]
Next: Long double, Previous: ANSI C Compliance, Up: Using the library [Index]
The inline keyword is not part of the original ANSI C standard
(C89) so the library does not export any inline function definitions
by default. Inline functions were introduced officially in the newer
C99 standard but most C89 compilers have also included inline as
an extension for a long time.
To allow the use of inline functions, the library provides optional
inline versions of performance-critical routines by conditional
compilation in the exported header files. The inline versions of these
functions can be included by defining the macro HAVE_INLINE
when compiling an application,
$ gcc -Wall -c -DHAVE_INLINE example.c
If you use autoconf this macro can be defined automatically. If
you do not define the macro HAVE_INLINE then the slower
non-inlined versions of the functions will be used instead.
By default, the actual form of the inline keyword is extern
inline, which is a gcc extension that eliminates unnecessary
function definitions. If the form extern inline causes
problems with other compilers a stricter autoconf test can be used,
see Autoconf Macros.
When compiling with gcc in C99 mode (gcc -std=c99) the
header files automatically switch to C99-compatible inline function
declarations instead of extern inline. With other C99
compilers, define the macro GSL_C99_INLINE to use these
declarations.
Next: Sparse Matrices Finding Maximum and Minimum Elements, Previous: Sparse Matrices Operations, Up: Sparse Matrices [Index]
This function returns the number of non-zero elements in m.
This function returns 1 if the matrices a and b are equal (by comparison of element values) and 0 otherwise. The matrices a and b must be in the same sparse storage format for comparison.
Next: Complex Generalized Hermitian-Definite Eigensystems, Previous: Real Nonsymmetric Matrices, Up: Eigensystems [Index]
The real generalized symmetric-definite eigenvalue problem is to find eigenvalues \lambda and eigenvectors x such that
A x = \lambda B x
where A and B are symmetric matrices, and B is positive-definite. This problem reduces to the standard symmetric eigenvalue problem by applying the Cholesky decomposition to B:
A x = \lambda B x
A x = \lambda L L^t x
( L^{-1} A L^{-t} ) L^t x = \lambda L^t x
Therefore, the problem becomes C y = \lambda y where C = L^{-1} A L^{-t} is symmetric, and y = L^t x. The standard symmetric eigensolver can be applied to the matrix C. The resulting eigenvectors are backtransformed to find the vectors of the original problem. The eigenvalues and eigenvectors of the generalized symmetric-definite eigenproblem are always real.
This function allocates a workspace for computing eigenvalues of n-by-n real generalized symmetric-definite eigensystems. The size of the workspace is O(2n).
This function frees the memory associated with the workspace w.
This function computes the eigenvalues of the real generalized symmetric-definite matrix pair (A, B), and stores them in eval, using the method outlined above. On output, B contains its Cholesky decomposition and A is destroyed.
This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real generalized symmetric-definite eigensystems. The size of the workspace is O(4n).
This function frees the memory associated with the workspace w.
This function computes the eigenvalues and eigenvectors of the real generalized symmetric-definite matrix pair (A, B), and stores them in eval and evec respectively. The computed eigenvectors are normalized to have unit magnitude. On output, B contains its Cholesky decomposition and A is destroyed.
Next: Complex Generalized Hermitian-Definite Eigensystems, Previous: Real Nonsymmetric Matrices, Up: Eigensystems [Index]
Next: Ntuple References and Further Reading, Previous: Histogramming ntuple values, Up: N-tuples [Index]
The following example programs demonstrate the use of ntuples in managing a large dataset. The first program creates a set of 10,000 simulated “events”, each with 3 associated values (x,y,z). These are generated from a Gaussian distribution with unit variance, for demonstration purposes, and written to the ntuple file test.dat.
#include <gsl/gsl_ntuple.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
struct data
{
double x;
double y;
double z;
};
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
struct data ntuple_row;
int i;
gsl_ntuple *ntuple
= gsl_ntuple_create ("test.dat", &ntuple_row,
sizeof (ntuple_row));
gsl_rng_env_setup ();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
for (i = 0; i < 10000; i++)
{
ntuple_row.x = gsl_ran_ugaussian (r);
ntuple_row.y = gsl_ran_ugaussian (r);
ntuple_row.z = gsl_ran_ugaussian (r);
gsl_ntuple_write (ntuple);
}
gsl_ntuple_close (ntuple);
gsl_rng_free (r);
return 0;
}
The next program analyses the ntuple data in the file test.dat. The analysis procedure is to compute the squared-magnitude of each event, E^2=x^2+y^2+z^2, and select only those which exceed a lower limit of 1.5. The selected events are then histogrammed using their E^2 values.
#include <math.h>
#include <gsl/gsl_ntuple.h>
#include <gsl/gsl_histogram.h>
struct data
{
double x;
double y;
double z;
};
int sel_func (void *ntuple_data, void *params);
double val_func (void *ntuple_data, void *params);
int
main (void)
{
struct data ntuple_row;
gsl_ntuple *ntuple
= gsl_ntuple_open ("test.dat", &ntuple_row,
sizeof (ntuple_row));
double lower = 1.5;
gsl_ntuple_select_fn S;
gsl_ntuple_value_fn V;
gsl_histogram *h = gsl_histogram_alloc (100);
gsl_histogram_set_ranges_uniform(h, 0.0, 10.0);
S.function = &sel_func;
S.params = &lower;
V.function = &val_func;
V.params = 0;
gsl_ntuple_project (h, ntuple, &V, &S);
gsl_histogram_fprintf (stdout, h, "%f", "%f");
gsl_histogram_free (h);
gsl_ntuple_close (ntuple);
return 0;
}
int
sel_func (void *ntuple_data, void *params)
{
struct data * data = (struct data *) ntuple_data;
double x, y, z, E2, scale;
scale = *(double *) params;
x = data->x;
y = data->y;
z = data->z;
E2 = x * x + y * y + z * z;
return E2 > scale;
}
double
val_func (void *ntuple_data, void *params)
{
(void)(params); /* avoid unused parameter warning */
struct data * data = (struct data *) ntuple_data;
double x, y, z;
x = data->x;
y = data->y;
z = data->z;
return x * x + y * y + z * z;
}
The following plot shows the distribution of the selected events. Note the cut-off at the lower bound.
Next: Ntuple References and Further Reading, Previous: Histogramming ntuple values, Up: N-tuples [Index]
Previous: Mixed-radix FFT routines for real data, Up: Fast Fourier Transforms [Index]
A good starting point for learning more about the FFT is the review article Fast Fourier Transforms: A Tutorial Review and A State of the Art by Duhamel and Vetterli,
To find out about the algorithms used in the GSL routines you may want to consult the document GSL FFT Algorithms (it is included in GSL, as doc/fftalgorithms.tex). This has general information on FFTs and explicit derivations of the implementation for each routine. There are also references to the relevant literature. For convenience some of the more important references are reproduced below.
There are several introductory books on the FFT with example programs, such as The Fast Fourier Transform by Brigham and DFT/FFT and Convolution Algorithms by Burrus and Parks,
Both these introductory books cover the radix-2 FFT in some detail. The mixed-radix algorithm at the heart of the FFTPACK routines is reviewed in Clive Temperton’s paper,
The derivation of FFTs for real-valued data is explained in the following two articles,
In 1979 the IEEE published a compendium of carefully-reviewed Fortran FFT programs in Programs for Digital Signal Processing. It is a useful reference for implementations of many different FFT algorithms,
For large-scale FFT work we recommend the use of the dedicated FFTW library by Frigo and Johnson. The FFTW library is self-optimizing—it automatically tunes itself for each hardware platform in order to achieve maximum performance. It is available under the GNU GPL.
The source code for FFTPACK is available from Netlib,
Previous: Mixed-radix FFT routines for real data, Up: Fast Fourier Transforms [Index]
Next: Laguerre Functions, Previous: Gegenbauer Functions, Up: Special Functions [Index]
Hypergeometric functions are described in Abramowitz & Stegun, Chapters 13 and 15. These functions are declared in the header file gsl_sf_hyperg.h.
These routines compute the hypergeometric function 0F1(c,x).
These routines compute the confluent hypergeometric function 1F1(m,n,x) = M(m,n,x) for integer parameters m, n.
These routines compute the confluent hypergeometric function 1F1(a,b,x) = M(a,b,x) for general parameters a, b.
These routines compute the confluent hypergeometric function U(m,n,x) for integer parameters m, n.
This routine computes the confluent hypergeometric function
U(m,n,x) for integer parameters m, n using the
gsl_sf_result_e10 type to return a result with extended range.
These routines compute the confluent hypergeometric function U(a,b,x).
This routine computes the confluent hypergeometric function
U(a,b,x) using the gsl_sf_result_e10 type to return a
result with extended range.
These routines compute the Gauss hypergeometric function 2F1(a,b,c,x) = F(a,b,c,x) for |x| < 1.
If the arguments (a,b,c,x) are too close to a singularity then
the function can return the error code GSL_EMAXITER when the
series approximation converges too slowly. This occurs in the region of
x=1, c - a - b = m for integer m.
These routines compute the Gauss hypergeometric function 2F1(a_R + i a_I, a_R - i a_I, c, x) with complex parameters for |x| < 1.
These routines compute the renormalized Gauss hypergeometric function 2F1(a,b,c,x) / \Gamma(c) for |x| < 1.
These routines compute the renormalized Gauss hypergeometric function 2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c) for |x| < 1.
These routines compute the hypergeometric function 2F0(a,b,x). The series representation is a divergent hypergeometric series. However, for x < 0 we have 2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)
Next: Laguerre Functions, Previous: Gegenbauer Functions, Up: Special Functions [Index]
Next: Initializing the B-splines solver, Up: Basis Splines [Index]
B-splines are commonly used as basis functions to fit smoothing curves to large data sets. To do this, the abscissa axis is broken up into some number of intervals, where the endpoints of each interval are called breakpoints. These breakpoints are then converted to knots by imposing various continuity and smoothness conditions at each interface. Given a nondecreasing knot vector t = {t_0, t_1, …, t_{n+k-1}}, the n basis splines of order k are defined by
B_(i,1)(x) = (1, t_i <= x < t_(i+1)
(0, else
B_(i,k)(x) = [(x - t_i)/(t_(i+k-1) - t_i)] B_(i,k-1)(x)
+ [(t_(i+k) - x)/(t_(i+k) - t_(i+1))] B_(i+1,k-1)(x)
for i = 0, …, n-1. The common case of cubic B-splines is given by k = 4. The above recurrence relation can be evaluated in a numerically stable way by the de Boor algorithm.
If we define appropriate knots on an interval [a,b] then the B-spline basis functions form a complete set on that interval. Therefore we can expand a smoothing function as
f(x) = \sum_i c_i B_(i,k)(x)
given enough (x_j, f(x_j)) data pairs. The coefficients c_i can be readily obtained from a least-squares fit.
gsl-ref-html-2.3/Random-Number-Distribution-Introduction.html 0000664 0001750 0001750 00000014127 13055414572 022436 0 ustar edd eddNext: The Gaussian Distribution, Up: Random Number Distributions [Index]
Continuous random number distributions are defined by a probability density function, p(x), such that the probability of x occurring in the infinitesimal range x to x+dx is p dx.
The cumulative distribution function for the lower tail P(x) is defined by the integral,
P(x) = \int_{-\infty}^{x} dx' p(x')
and gives the probability of a variate taking a value less than x.
The cumulative distribution function for the upper tail Q(x) is defined by the integral,
Q(x) = \int_{x}^{+\infty} dx' p(x')
and gives the probability of a variate taking a value greater than x.
The upper and lower cumulative distribution functions are related by P(x) + Q(x) = 1 and satisfy 0 <= P(x) <= 1, 0 <= Q(x) <= 1.
The inverse cumulative distributions, x=P^{-1}(P) and x=Q^{-1}(Q) give the values of x which correspond to a specific value of P or Q. They can be used to find confidence limits from probability values.
For discrete distributions the probability of sampling the integer value k is given by p(k), where \sum_k p(k) = 1. The cumulative distribution for the lower tail P(k) of a discrete distribution is defined as,
P(k) = \sum_{i <= k} p(i)
where the sum is over the allowed range of the distribution less than or equal to k.
The cumulative distribution for the upper tail of a discrete distribution Q(k) is defined as
Q(k) = \sum_{i > k} p(i)
giving the sum of probabilities for all values greater than k. These two definitions satisfy the identity P(k)+Q(k)=1.
If the range of the distribution is 1 to n inclusive then P(n)=1, Q(n)=0 while P(1) = p(1), Q(1)=1-p(1).
Next: The Gaussian Distribution, Up: Random Number Distributions [Index]
Next: DWT Initialization, Up: Wavelet Transforms [Index]
The continuous wavelet transform and its inverse are defined by the relations,
w(s,\tau) = \int f(t) * \psi^*_{s,\tau}(t) dt
and,
f(t) = \int \int_{-\infty}^\infty w(s, \tau) * \psi_{s,\tau}(t) d\tau ds
where the basis functions \psi_{s,\tau} are obtained by scaling and translation from a single function, referred to as the mother wavelet.
The discrete version of the wavelet transform acts on equally-spaced samples, with fixed scaling and translation steps (s, \tau). The frequency and time axes are sampled dyadically on scales of 2^j through a level parameter j. The resulting family of functions {\psi_{j,n}} constitutes an orthonormal basis for square-integrable signals.
The discrete wavelet transform is an O(N) algorithm, and is also referred to as the fast wavelet transform.
gsl-ref-html-2.3/Irregular-Bessel-Functions-_002d-Fractional-Order.html 0000664 0001750 0001750 00000011077 13055414521 023677 0 ustar edd eddNext: Regular Modified Bessel Functions - Fractional Order, Previous: Regular Bessel Function - Fractional Order, Up: Bessel Functions [Index]
These routines compute the irregular cylindrical Bessel function of fractional order \nu, Y_\nu(x).
Next: Sparse Matrices Allocation, Up: Sparse Matrices [Index]
These routines provide support for constructing and manipulating
sparse matrices in GSL, using an API similar to gsl_matrix.
The basic structure is called gsl_spmatrix. There are
three supported storage formats for sparse matrices: the triplet,
compressed column storage (CCS) and compressed row storage (CRS)
formats. The triplet format stores triplets (i,j,x) for each
non-zero element of the matrix. This notation means that the
(i,j) element of the matrix A
is A_{ij} = x. Compressed column storage stores each column of
non-zero values in the sparse matrix in a continuous memory block, keeping
pointers to the beginning of each column in that memory block, and storing
the row indices of each non-zero element. Compressed row storage stores
each row of non-zero values in a continuous memory block, keeping pointers
to the beginning of each row in the block and storing the column indices
of each non-zero element. The triplet format is ideal
for adding elements to the sparse matrix structure while it is being
constructed, while the compressed storage formats are better suited for
matrix-matrix multiplication or linear solvers.
The gsl_spmatrix structure is defined as
typedef struct
{
size_t size1;
size_t size2;
size_t *i;
double *data;
size_t *p;
size_t nzmax;
size_t nz;
gsl_spmatrix_tree *tree_data;
void *work;
size_t sptype;
} gsl_spmatrix;
This defines a size1-by-size2 sparse matrix. The number of non-zero elements currently in the matrix is given by nz. For the triplet representation, i, p, and data are arrays of size nz which contain the row indices, column indices, and element value, respectively. So if data[k] = A(i,j), then i = i[k] and j = p[k].
For compressed column storage, i and data are arrays of size nz containing the row indices and element values, identical to the triplet case. p is an array of size size2 + 1 where p[j] points to the index in data of the start of column j. Thus, if data[k] = A(i,j), then i = i[k] and p[j] <= k < p[j+1].
For compressed row storage, i and data are arrays of size nz containing the column indices and element values, identical to the triplet case. p is an array of size size1 + 1 where p[i] points to the index in data of the start of row i. Thus, if data[k] = A(i,j), then j = i[k] and p[i] <= k < p[i+1].
The parameter tree_data is a binary tree structure used in the triplet representation, specifically a balanced AVL tree. This speeds up element searches and duplicate detection during the matrix assembly process. The parameter work is additional workspace needed for various operations like converting from triplet to compressed storage. sptype indicates the type of storage format being used (triplet, CCS or CRS).
The compressed storage format defined above makes it very simple to interface with sophisticated external linear solver libraries which accept compressed storage input. The user can simply pass the arrays i, p, and data as the inputs to external libraries.
Next: Sparse Matrices Allocation, Up: Sparse Matrices [Index]
Next: Nonlinear Least-Squares Initialization, Previous: Nonlinear Least-Squares Weighted Overview, Up: Nonlinear Least-Squares Fitting [Index]
The user can tune nearly all aspects of the iteration at allocation
time. For the gsl_multifit_nlinear interface, the user may
modify the gsl_multifit_nlinear_parameters structure, which is
defined as follows:
typedef struct
{
const gsl_multifit_nlinear_trs *trs; /* trust region subproblem method */
const gsl_multifit_nlinear_scale *scale; /* scaling method */
const gsl_multifit_nlinear_solver *solver; /* solver method */
gsl_multifit_nlinear_fdtype fdtype; /* finite difference method */
double factor_up; /* factor for increasing trust radius */
double factor_down; /* factor for decreasing trust radius */
double avmax; /* max allowed |a|/|v| */
double h_df; /* step size for finite difference Jacobian */
double h_fvv; /* step size for finite difference fvv */
} gsl_multifit_nlinear_parameters;
For the gsl_multilarge_nlinear interface, the user may
modify the gsl_multilarge_nlinear_parameters structure, which is
defined as follows:
typedef struct
{
const gsl_multilarge_nlinear_trs *trs; /* trust region subproblem method */
const gsl_multilarge_nlinear_scale *scale; /* scaling method */
const gsl_multilarge_nlinear_solver *solver; /* solver method */
gsl_multilarge_nlinear_fdtype fdtype; /* finite difference method */
double factor_up; /* factor for increasing trust radius */
double factor_down; /* factor for decreasing trust radius */
double avmax; /* max allowed |a|/|v| */
double h_df; /* step size for finite difference Jacobian */
double h_fvv; /* step size for finite difference fvv */
size_t max_iter; /* maximum iterations for trs method */
double tol; /* tolerance for solving trs */
} gsl_multilarge_nlinear_parameters;
Each of these parameters is discussed in further detail below.
This parameter determines the method used to solve the trust region subproblem, and may be selected from the following choices,
This selects the Levenberg-Marquardt algorithm.
This selects the Levenberg-Marquardt algorithm with geodesic acceleration.
This selects the dogleg algorithm.
This selects the double dogleg algorithm.
This parameter determines the diagonal scaling matrix D and may be selected from the following choices,
This damping strategy was suggested by Moré, and corresponds to D^T D = max(diag(J^T J)), in other words the maximum elements of diag(J^T J) encountered thus far in the iteration. This choice of D makes the problem scale-invariant, so that if the model parameters x_i are each scaled by an arbitrary constant, \tilde{x}_i = a_i x_i, then the sequence of iterates produced by the algorithm would be unchanged. This method can work very well in cases where the model parameters have widely different scales (ie: if some parameters are measured in nanometers, while others are measured in degrees Kelvin). This strategy has been proven effective on a large class of problems and so it is the library default, but it may not be the best choice for all problems.
This damping strategy was originally suggested by Levenberg, and corresponds to D^T D = I. This method has also proven effective on a large class of problems, but is not scale-invariant. However, some authors (e.g. Transtrum and Sethna 2012) argue that this choice is better for problems which are susceptible to parameter evaporation (ie: parameters go to infinity)
This damping strategy was suggested by Marquardt, and corresponds to D^T D = diag(J^T J). This method is scale-invariant, but it is generally considered inferior to both the Levenberg and Moré strategies, though may work well on certain classes of problems.
Solving the trust region subproblem on each iteration almost always requires the solution of the following linear least squares system
[J; sqrt(mu) D] \delta = - [f; 0]
The solver parameter determines how the system is solved and can be selected from the following choices:
This method solves the system using a rank revealing QR decomposition of the Jacobian J. This method will produce reliable solutions in cases where the Jacobian is rank deficient or near-singular but does require about twice as many operations as the Cholesky method discussed below.
This method solves the alternate normal equations problem
( J^T J + \mu D^T D ) \delta = -J^T f
by using a Cholesky decomposition of the matrix J^T J + \mu D^T D. This method is faster than the QR approach, however it is susceptible to numerical instabilities if the Jacobian matrix is rank deficient or near-singular. In these cases, an attempt is made to reduce the condition number of the matrix using Jacobi preconditioning, but for highly ill-conditioned problems the QR approach is better. If it is known that the Jacobian matrix is well conditioned, this method is accurate and will perform faster than the QR approach.
This parameter specifies whether to use forward or centered differences when approximating the Jacobian. This is only used when an analytic Jacobian is not provided to the solver. This parameter may be set to one of the following choices.
This specifies a forward finite difference to approximate the Jacobian matrix. The Jacobian matrix will be calculated as
J_ij = 1 / \Delta_j ( f_i(x + \Delta_j e_j) - f_i(x) )
where \Delta_j = h |x_j| and e_j is the standard
jth Cartesian unit basis vector so that
x + \Delta_j e_j represents a small (forward) perturbation of
the jth parameter by an amount \Delta_j. The perturbation
\Delta_j is proportional to the current value |x_j| which
helps to calculate an accurate Jacobian when the various parameters have
different scale sizes. The value of h is specified by the h_df
parameter. The accuracy of this method is O(h), and evaluating this
matrix requires an additional p function evaluations.
This specifies a centered finite difference to approximate the Jacobian matrix. The Jacobian matrix will be calculated as
J_ij = 1 / \Delta_j ( f_i(x + 1/2 \Delta_j e_j) - f_i(x - 1/2 \Delta_j e_j) )
See above for a description of \Delta_j. The accuracy of this method is O(h^2), but evaluating this matrix requires an additional 2p function evaluations.
When a step is accepted, the trust region radius will be increased by this factor. The default value is 3.
When a step is rejected, the trust region radius will be decreased by this factor. The default value is 2.
When using geodesic acceleration to solve a nonlinear least squares problem, an important parameter to monitor is the ratio of the acceleration term to the velocity term,
|a| / |v|
If this ratio is small, it means the acceleration correction is contributing very little to the step. This could be because the problem is not “nonlinear” enough to benefit from the acceleration. If the ratio is large (> 1) it means that the acceleration is larger than the velocity, which shouldn’t happen since the step represents a truncated series and so the second order term a should be smaller than the first order term v to guarantee convergence. Therefore any steps with a ratio larger than the parameter avmax are rejected. avmax is set to 0.75 by default. For problems which experience difficulty converging, this threshold could be lowered.
This parameter specifies the step size for approximating the
Jacobian matrix with finite differences. It is set to
\sqrt{\epsilon} by default, where \epsilon
is GSL_DBL_EPSILON.
When using geodesic acceleration, the user must either supply a function to calculate f_{vv}(x) or the library can estimate this second directional derivative using a finite difference method. When using finite differences, the library must calculate f(x + h v) where h represents a small step in the velocity direction. The parameter h_fvv defines this step size and is set to 0.02 by default.
Next: Nonlinear Least-Squares Initialization, Previous: Nonlinear Least-Squares Weighted Overview, Up: Nonlinear Least-Squares Fitting [Index]
Next: Introduction, Previous: (dir), Up: (dir) [Index]
This file documents the GNU Scientific Library (GSL), a collection of numerical routines for scientific computing. It corresponds to release 2.3 of the library. Please report any errors in this manual to bug-gsl@gnu.org.
More information about GSL can be found at the project homepage, http://www.gnu.org/software/gsl/.
Printed copies of this manual can be purchased from Network Theory Ltd at http://www.network-theory.co.uk/gsl/manual/. The money raised from sales of the manual helps support the development of GSL.
A Japanese translation of this manual is available from the GSL project homepage thanks to Daisuke Tominaga.
Copyright © 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016 The GSL Team.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with the Invariant Sections being “GNU General Public License” and “Free Software Needs Free Documentation”, the Front-Cover text being “A GNU Manual”, and with the Back-Cover Text being (a) (see below). A copy of the license is included in the section entitled “GNU Free Documentation License”.
(a) The Back-Cover Text is: “You have the freedom to copy and modify this GNU Manual.”
Next: Introduction, Previous: (dir), Up: (dir) [Index]
Next: Light and Illumination, Previous: Pressure, Up: Physical Constants [Index]
GSL_CONST_MKSA_POISEThe dynamic viscosity of 1 poise.
GSL_CONST_MKSA_STOKESThe kinematic viscosity of 1 stokes.
Previous: Running Statistics Example programs, Up: Running Statistics [Index]
The algorithm used to dynamically estimate p-quantiles is described in the paper,
Next: Fermi-Dirac Function, Previous: Exponential Functions, Up: Special Functions [Index]
Information on the exponential integrals can be found in Abramowitz & Stegun, Chapter 5. These functions are declared in the header file gsl_sf_expint.h.
| • Exponential Integral: | ||
| • Ei(x): | ||
| • Hyperbolic Integrals: | ||
| • Ei_3(x): | ||
| • Trigonometric Integrals: | ||
| • Arctangent Integral: |
Next: Large Dense Linear Systems Routines, Previous: Large Dense Linear Systems TSQR, Up: Large Dense Linear Systems [Index]
The typical steps required to solve large regularized linear least squares problems are as follows:
Next: Testing for Odd and Even Numbers, Previous: Small integer powers, Up: Mathematical Functions [Index]
This macro returns the sign of x. It is defined as ((x) >= 0
? 1 : -1). Note that with this definition the sign of zero is positive
(regardless of its IEEE sign bit).
Next: Complementary Error Function, Up: Error Functions [Index]
These routines compute the error function erf(x), where erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2).
Next: Vectors and Matrices, Previous: Polynomials, Up: Top [Index]
This chapter describes the GSL special function library. The library includes routines for calculating the values of Airy functions, Bessel functions, Clausen functions, Coulomb wave functions, Coupling coefficients, the Dawson function, Debye functions, Dilogarithms, Elliptic integrals, Jacobi elliptic functions, Error functions, Exponential integrals, Fermi-Dirac functions, Gamma functions, Gegenbauer functions, Hypergeometric functions, Laguerre functions, Legendre functions and Spherical Harmonics, the Psi (Digamma) Function, Synchrotron functions, Transport functions, Trigonometric functions and Zeta functions. Each routine also computes an estimate of the numerical error in the calculated value of the function.
The functions in this chapter are declared in individual header files, such as gsl_sf_airy.h, gsl_sf_bessel.h, etc. The complete set of header files can be included using the file gsl_sf.h.
Next: Vectors and Matrices, Previous: Polynomials, Up: Top [Index]
Next: Regular Spherical Bessel Functions, Previous: Regular Modified Cylindrical Bessel Functions, Up: Bessel Functions [Index]
These routines compute the irregular modified cylindrical Bessel function of zeroth order, K_0(x), for x > 0.
These routines compute the irregular modified cylindrical Bessel function of first order, K_1(x), for x > 0.
These routines compute the irregular modified cylindrical Bessel function of order n, K_n(x), for x > 0.
This routine computes the values of the irregular modified cylindrical Bessel functions K_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The start of the range nmin must be positive or zero. The domain of the function is x>0. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
These routines compute the scaled irregular modified cylindrical Bessel function of zeroth order \exp(x) K_0(x) for x>0.
These routines compute the scaled irregular modified cylindrical Bessel function of first order \exp(x) K_1(x) for x>0.
These routines compute the scaled irregular modified cylindrical Bessel function of order n, \exp(x) K_n(x), for x>0.
This routine computes the values of the scaled irregular cylindrical Bessel functions \exp(x) K_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The start of the range nmin must be positive or zero. The domain of the function is x>0. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
Next: Regular Spherical Bessel Functions, Previous: Regular Modified Cylindrical Bessel Functions, Up: Bessel Functions [Index]
Next: Legendre Form of Incomplete Elliptic Integrals, Previous: Definition of Carlson Forms, Up: Elliptic Integrals [Index]
These routines compute the complete elliptic integral K(k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.
These routines compute the complete elliptic integral E(k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.
These routines compute the complete elliptic integral \Pi(k,n) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of sign n \to -n.
Next: Minimization Stopping Parameters, Previous: Providing the function to minimize, Up: One dimensional Minimization [Index]
The following functions drive the iteration of each algorithm. Each function performs one iteration to update the state of any minimizer of the corresponding type. The same functions work for all minimizers so that different methods can be substituted at runtime without modifications to the code.
This function performs a single iteration of the minimizer s. If the iteration encounters an unexpected problem then an error code will be returned,
GSL_EBADFUNCthe iteration encountered a singular point where the function evaluated
to Inf or NaN.
GSL_FAILUREthe algorithm could not improve the current best approximation or bounding interval.
The minimizer maintains a current best estimate of the position of the minimum at all times, and the current interval bounding the minimum. This information can be accessed with the following auxiliary functions,
This function returns the current estimate of the position of the minimum for the minimizer s.
These functions return the current upper and lower bound of the interval for the minimizer s.
These functions return the value of the function at the current estimate of the minimum and at the upper and lower bounds of the interval for the minimizer s.
Next: Real Generalized Symmetric-Definite Eigensystems, Previous: Complex Hermitian Matrices, Up: Eigensystems [Index]
The solution of the real nonsymmetric eigensystem problem for a matrix A involves computing the Schur decomposition
A = Z T Z^T
where Z is an orthogonal matrix of Schur vectors and T, the Schur form, is quasi upper triangular with diagonal 1-by-1 blocks which are real eigenvalues of A, and diagonal 2-by-2 blocks whose eigenvalues are complex conjugate eigenvalues of A. The algorithm used is the double-shift Francis method.
This function allocates a workspace for computing eigenvalues of n-by-n real nonsymmetric matrices. The size of the workspace is O(2n).
This function frees the memory associated with the workspace w.
This function sets some parameters which determine how the eigenvalue
problem is solved in subsequent calls to gsl_eigen_nonsymm.
If compute_t is set to 1, the full Schur form T will be
computed by gsl_eigen_nonsymm. If it is set to 0,
T will not be computed (this is the default setting). Computing
the full Schur form T requires approximately 1.5–2 times the
number of flops.
If balance is set to 1, a balancing transformation is applied
to the matrix prior to computing eigenvalues. This transformation is
designed to make the rows and columns of the matrix have comparable
norms, and can result in more accurate eigenvalues for matrices
whose entries vary widely in magnitude. See Balancing for more
information. Note that the balancing transformation does not preserve
the orthogonality of the Schur vectors, so if you wish to compute the
Schur vectors with gsl_eigen_nonsymm_Z you will obtain the Schur
vectors of the balanced matrix instead of the original matrix. The
relationship will be
T = Q^t D^(-1) A D Q
where Q is the matrix of Schur vectors for the balanced matrix, and
D is the balancing transformation. Then gsl_eigen_nonsymm_Z
will compute a matrix Z which satisfies
T = Z^(-1) A Z
with Z = D Q. Note that Z will not be orthogonal. For this reason, balancing is not performed by default.
This function computes the eigenvalues of the real nonsymmetric matrix
A and stores them in the vector eval. If T is
desired, it is stored in the upper portion of A on output.
Otherwise, on output, the diagonal of A will contain the
1-by-1 real eigenvalues and 2-by-2
complex conjugate eigenvalue systems, and the rest of A is
destroyed. In rare cases, this function may fail to find all
eigenvalues. If this happens, an error code is returned
and the number of converged eigenvalues is stored in w->n_evals.
The converged eigenvalues are stored in the beginning of eval.
This function is identical to gsl_eigen_nonsymm except that it also
computes the Schur vectors and stores them into Z.
This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real nonsymmetric matrices. The size of the workspace is O(5n).
This function frees the memory associated with the workspace w.
This function sets parameters which determine how the eigenvalue
problem is solved in subsequent calls to gsl_eigen_nonsymmv.
If balance is set to 1, a balancing transformation is applied
to the matrix. See gsl_eigen_nonsymm_params for more information.
Balancing is turned off by default since it does not preserve the
orthogonality of the Schur vectors.
This function computes eigenvalues and right eigenvectors of the
n-by-n real nonsymmetric matrix A. It first calls
gsl_eigen_nonsymm to compute the eigenvalues, Schur form T, and
Schur vectors. Then it finds eigenvectors of T and backtransforms
them using the Schur vectors. The Schur vectors are destroyed in the
process, but can be saved by using gsl_eigen_nonsymmv_Z. The
computed eigenvectors are normalized to have unit magnitude. On
output, the upper portion of A contains the Schur form
T. If gsl_eigen_nonsymm fails, no eigenvectors are
computed, and an error code is returned.
This function is identical to gsl_eigen_nonsymmv except that it also saves
the Schur vectors into Z.
Next: Real Generalized Symmetric-Definite Eigensystems, Previous: Complex Hermitian Matrices, Up: Eigensystems [Index]
Next: The Pareto Distribution, Previous: The Beta Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the logistic distribution. The distribution function is,
p(x) dx = { \exp(-x/a) \over a (1 + \exp(-x/a))^2 } dx
for -\infty < x < +\infty.
This function computes the probability density p(x) at x for a logistic distribution with scale parameter a, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the logistic distribution with scale parameter a.
Next: Permutation functions, Previous: Accessing permutation elements, Up: Permutations [Index]
This function returns the size of the permutation p.
This function returns a pointer to the array of elements in the permutation p.
This function checks that the permutation p is valid. The n elements should contain each of the numbers 0 to n-1 once and only once.
Next: Incomplete Gamma Functions, Previous: Factorials, Up: Gamma and Beta Functions [Index]
These routines compute the Pochhammer symbol (a)_x = \Gamma(a + x)/\Gamma(a). The Pochhammer symbol is also known as the Apell symbol and sometimes written as (a,x). When a and a+x are negative integers or zero, the limiting value of the ratio is returned.
These routines compute the logarithm of the Pochhammer symbol, \log((a)_x) = \log(\Gamma(a + x)/\Gamma(a)).
These routines compute the sign of the Pochhammer symbol and the logarithm of its magnitude. The computed parameters are result = \log(|(a)_x|) with a corresponding error term, and sgn = \sgn((a)_x) where (a)_x = \Gamma(a + x)/\Gamma(a).
These routines compute the relative Pochhammer symbol ((a)_x - 1)/x where (a)_x = \Gamma(a + x)/\Gamma(a).
Next: Updating and accessing histogram elements, Previous: Histogram allocation, Up: Histograms [Index]
This function copies the histogram src into the pre-existing histogram dest, making dest into an exact copy of src. The two histograms must be of the same size.
This function returns a pointer to a newly created histogram which is an exact copy of the histogram src.
Next: Legendre Form of Complete Elliptic Integrals, Previous: Definition of Legendre Forms, Up: Elliptic Integrals [Index]
The Carlson symmetric forms of elliptical integrals RC(x,y), RD(x,y,z), RF(x,y,z) and RJ(x,y,z,p) are defined by,
RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)
RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)
RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)
RJ(x,y,z,p) = 3/2 \int_0^\infty dt
(t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)
Next: The Bivariate Gaussian Distribution, Previous: The Gaussian Distribution, Up: Random Number Distributions [Index]
This function provides random variates from the upper tail of a Gaussian distribution with standard deviation sigma. The values returned are larger than the lower limit a, which must be positive. The method is based on Marsaglia’s famous rectangle-wedge-tail algorithm (Ann. Math. Stat. 32, 894–899 (1961)), with this aspect explained in Knuth, v2, 3rd ed, p139,586 (exercise 11).
The probability distribution for Gaussian tail random variates is,
p(x) dx = {1 \over N(a;\sigma) \sqrt{2 \pi \sigma^2}} \exp (- x^2/(2 \sigma^2)) dx
for x > a where N(a;\sigma) is the normalization constant,
N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).
This function computes the probability density p(x) at x for a Gaussian tail distribution with standard deviation sigma and lower limit a, using the formula given above.
These functions compute results for the tail of a unit Gaussian distribution. They are equivalent to the functions above with a standard deviation of one, sigma = 1.
Next: Error Functions, Previous: Elliptic Integrals, Up: Special Functions [Index]
The Jacobian Elliptic functions are defined in Abramowitz & Stegun, Chapter 16. The functions are declared in the header file gsl_sf_elljac.h.
This function computes the Jacobian elliptic functions sn(u|m), cn(u|m), dn(u|m) by descending Landen transformations.
Previous: Restriction Functions, Up: Trigonometric Functions [Index]
This routine computes the sine of an angle x with an associated absolute error dx, \sin(x \pm dx). Note that this function is provided in the error-handling form only since its purpose is to compute the propagated error.
This routine computes the cosine of an angle x with an associated absolute error dx, \cos(x \pm dx). Note that this function is provided in the error-handling form only since its purpose is to compute the propagated error.
Next: 1D Higher-level Interface, Previous: 1D Index Look-up and Acceleration, Up: Interpolation [Index]
These functions return the interpolated value of y for a given
point x, using the interpolation object interp, data
arrays xa and ya and the accelerator acc. When
x is outside the range of xa, the error code
GSL_EDOM is returned with a value of GSL_NAN for
y.
These functions return the derivative d of an interpolated function for a given point x, using the interpolation object interp, data arrays xa and ya and the accelerator acc.
These functions return the second derivative d2 of an interpolated function for a given point x, using the interpolation object interp, data arrays xa and ya and the accelerator acc.
These functions return the numerical integral result of an interpolated function over the range [a, b], using the interpolation object interp, data arrays xa and ya and the accelerator acc.
Next: 1D Higher-level Interface, Previous: 1D Index Look-up and Acceleration, Up: Interpolation [Index]
Next: Fitting robust linear regression example, Previous: Fitting regularized linear regression example 1, Up: Fitting Examples [Index]
The following example program minimizes the cost function
||y - X c||^2 + \lambda^2 ||x||^2
where X is the 10-by-8 Hilbert matrix whose entries are given by
X_{ij} = 1 / (i + j - 1)
and the right hand side vector is given by y = [1,-1,1,-1,1,-1,1,-1,1,-1]^T. Solutions are computed for \lambda = 0 (unregularized) as well as for optimal parameters \lambda chosen by analyzing the L-curve and GCV curve.
Here is the program output:
matrix condition number = 3.565872e+09 === Unregularized fit === residual norm = 2.15376 solution norm = 2.92217e+09 chisq/dof = 2.31934 === Regularized fit (L-curve) === optimal lambda: 7.11407e-07 residual norm = 2.60386 solution norm = 424507 chisq/dof = 3.43565 === Regularized fit (GCV) === optimal lambda: 1.72278 residual norm = 3.1375 solution norm = 0.139357 chisq/dof = 4.95076
Here we see the unregularized solution results in a large solution norm due to the ill-conditioned matrix. The L-curve solution finds a small value of \lambda = 7.11e-7 which still results in a badly conditioned system and a large solution norm. The GCV method finds a parameter \lambda = 1.72 which results in a well-conditioned system and small solution norm.
The L-curve and its computed corner, as well as the GCV curve and its minimum are plotted below.
The program is given below.
#include <gsl/gsl_math.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_multifit.h>
#include <gsl/gsl_blas.h>
static int
hilbert_matrix(gsl_matrix * m)
{
const size_t N = m->size1;
const size_t M = m->size2;
size_t i, j;
for (i = 0; i < N; i++)
{
for (j = 0; j < M; j++)
{
gsl_matrix_set(m, i, j, 1.0/(i+j+1.0));
}
}
return GSL_SUCCESS;
}
int
main()
{
const size_t n = 10; /* number of observations */
const size_t p = 8; /* number of model parameters */
size_t i;
gsl_matrix *X = gsl_matrix_alloc(n, p);
gsl_vector *y = gsl_vector_alloc(n);
/* construct Hilbert matrix and rhs vector */
hilbert_matrix(X);
{
double val = 1.0;
for (i = 0; i < n; ++i)
{
gsl_vector_set(y, i, val);
val *= -1.0;
}
}
{
const size_t npoints = 200; /* number of points on L-curve and GCV curve */
gsl_multifit_linear_workspace *w =
gsl_multifit_linear_alloc(n, p);
gsl_vector *c = gsl_vector_alloc(p); /* OLS solution */
gsl_vector *c_lcurve = gsl_vector_alloc(p); /* regularized solution (L-curve) */
gsl_vector *c_gcv = gsl_vector_alloc(p); /* regularized solution (GCV) */
gsl_vector *reg_param = gsl_vector_alloc(npoints);
gsl_vector *rho = gsl_vector_alloc(npoints); /* residual norms */
gsl_vector *eta = gsl_vector_alloc(npoints); /* solution norms */
gsl_vector *G = gsl_vector_alloc(npoints); /* GCV function values */
double lambda_l; /* optimal regularization parameter (L-curve) */
double lambda_gcv; /* optimal regularization parameter (GCV) */
double G_gcv; /* G(lambda_gcv) */
size_t reg_idx; /* index of optimal lambda */
double rcond; /* reciprocal condition number of X */
double chisq, rnorm, snorm;
/* compute SVD of X */
gsl_multifit_linear_svd(X, w);
rcond = gsl_multifit_linear_rcond(w);
fprintf(stderr, "matrix condition number = %e\n", 1.0 / rcond);
/* unregularized (standard) least squares fit, lambda = 0 */
gsl_multifit_linear_solve(0.0, X, y, c, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0);
fprintf(stderr, "=== Unregularized fit ===\n");
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));
/* calculate L-curve and find its corner */
gsl_multifit_linear_lcurve(y, reg_param, rho, eta, w);
gsl_multifit_linear_lcorner(rho, eta, ®_idx);
/* store optimal regularization parameter */
lambda_l = gsl_vector_get(reg_param, reg_idx);
/* regularize with lambda_l */
gsl_multifit_linear_solve(lambda_l, X, y, c_lcurve, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);
fprintf(stderr, "=== Regularized fit (L-curve) ===\n");
fprintf(stderr, "optimal lambda: %g\n", lambda_l);
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));
/* calculate GCV curve and find its minimum */
gsl_multifit_linear_gcv(y, reg_param, G, &lambda_gcv, &G_gcv, w);
/* regularize with lambda_gcv */
gsl_multifit_linear_solve(lambda_gcv, X, y, c_gcv, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0) + pow(lambda_gcv * snorm, 2.0);
fprintf(stderr, "=== Regularized fit (GCV) ===\n");
fprintf(stderr, "optimal lambda: %g\n", lambda_gcv);
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));
/* output L-curve and GCV curve */
for (i = 0; i < npoints; ++i)
{
printf("%e %e %e %e\n",
gsl_vector_get(reg_param, i),
gsl_vector_get(rho, i),
gsl_vector_get(eta, i),
gsl_vector_get(G, i));
}
/* output L-curve corner point */
printf("\n\n%f %f\n",
gsl_vector_get(rho, reg_idx),
gsl_vector_get(eta, reg_idx));
/* output GCV curve corner minimum */
printf("\n\n%e %e\n",
lambda_gcv,
G_gcv);
gsl_multifit_linear_free(w);
gsl_vector_free(c);
gsl_vector_free(c_lcurve);
gsl_vector_free(reg_param);
gsl_vector_free(rho);
gsl_vector_free(eta);
gsl_vector_free(G);
}
gsl_matrix_free(X);
gsl_vector_free(y);
return 0;
}
Next: Fitting robust linear regression example, Previous: Fitting regularized linear regression example 1, Up: Fitting Examples [Index]
Previous: Monte Carlo Examples, Up: Monte Carlo Integration [Index]
The MISER algorithm is described in the following article by Press and Farrar,
The VEGAS algorithm is described in the following papers,
Next: Simulated Annealing References and Further Reading, Previous: Simulated Annealing functions, Up: Simulated Annealing [Index]
The simulated annealing package is clumsy, and it has to be because it is written in C, for C callers, and tries to be polymorphic at the same time. But here we provide some examples which can be pasted into your application with little change and should make things easier.
| • Trivial example: | ||
| • Traveling Salesman Problem: |
Next: Tridiagonal Systems, Previous: Householder Transformations, Up: Linear Algebra [Index]
This function solves the system A x = b directly using Householder transformations. On output the solution is stored in x and b is not modified. The matrix A is destroyed by the Householder transformations.
This function solves the system A x = b in-place using Householder transformations. On input x should contain the right-hand side b, which is replaced by the solution on output. The matrix A is destroyed by the Householder transformations.
Previous: Example statistical programs, Up: Statistics [Index]
The standard reference for almost any topic in statistics is the multi-volume Advanced Theory of Statistics by Kendall and Stuart.
Many statistical concepts can be more easily understood by a Bayesian approach. The following book by Gelman, Carlin, Stern and Rubin gives a comprehensive coverage of the subject.
For physicists the Particle Data Group provides useful reviews of Probability and Statistics in the “Mathematical Tools” section of its Annual Review of Particle Physics.
The Review of Particle Physics is available online at the website http://pdg.lbl.gov/.
gsl-ref-html-2.3/Regular-Modified-Cylindrical-Bessel-Functions.html 0000664 0001750 0001750 00000022101 13055414520 023347 0 ustar edd eddNext: Irregular Modified Cylindrical Bessel Functions, Previous: Irregular Cylindrical Bessel Functions, Up: Bessel Functions [Index]
These routines compute the regular modified cylindrical Bessel function of zeroth order, I_0(x).
These routines compute the regular modified cylindrical Bessel function of first order, I_1(x).
These routines compute the regular modified cylindrical Bessel function of order n, I_n(x).
This routine computes the values of the regular modified cylindrical Bessel functions I_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The start of the range nmin must be positive or zero. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
These routines compute the scaled regular modified cylindrical Bessel function of zeroth order \exp(-|x|) I_0(x).
These routines compute the scaled regular modified cylindrical Bessel function of first order \exp(-|x|) I_1(x).
These routines compute the scaled regular modified cylindrical Bessel function of order n, \exp(-|x|) I_n(x)
This routine computes the values of the scaled regular cylindrical Bessel functions \exp(-|x|) I_n(x) for n from nmin to nmax inclusive, storing the results in the array result_array. The start of the range nmin must be positive or zero. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
Next: Irregular Modified Cylindrical Bessel Functions, Previous: Irregular Cylindrical Bessel Functions, Up: Bessel Functions [Index]
Next: Speed and Nautical Units, Previous: Measurement of Time, Up: Physical Constants [Index]
GSL_CONST_MKSA_INCHThe length of 1 inch.
GSL_CONST_MKSA_FOOTThe length of 1 foot.
GSL_CONST_MKSA_YARDThe length of 1 yard.
GSL_CONST_MKSA_MILEThe length of 1 mile.
GSL_CONST_MKSA_MILThe length of 1 mil (1/1000th of an inch).
Previous: Minimization Examples, Up: One dimensional Minimization [Index]
Further information on Brent’s algorithm is available in the following book,
Next: Trigonometric Functions With Error Estimates, Previous: Conversion Functions, Up: Trigonometric Functions [Index]
These routines force the angle theta to lie in the range (-\pi,\pi].
Note that the mathematical value of \pi is slightly greater
than M_PI, so the machine numbers M_PI and -M_PI
are included in the range.
These routines force the angle theta to lie in the range [0, 2\pi).
Note that the mathematical value of 2\pi is slightly greater
than 2*M_PI, so the machine number 2*M_PI is included in
the range.
Next: Quasi-random number references, Previous: Quasi-random number generator algorithms, Up: Quasi-Random Sequences [Index]
The following program prints the first 1024 points of the 2-dimensional Sobol sequence.
#include <stdio.h>
#include <gsl/gsl_qrng.h>
int
main (void)
{
int i;
gsl_qrng * q = gsl_qrng_alloc (gsl_qrng_sobol, 2);
for (i = 0; i < 1024; i++)
{
double v[2];
gsl_qrng_get (q, v);
printf ("%.5f %.5f\n", v[0], v[1]);
}
gsl_qrng_free (q);
return 0;
}
Here is the output from the program,
$ ./a.out 0.50000 0.50000 0.75000 0.25000 0.25000 0.75000 0.37500 0.37500 0.87500 0.87500 0.62500 0.12500 0.12500 0.62500 ....
It can be seen that successive points progressively fill-in the spaces between previous points.
gsl-ref-html-2.3/Legendre-Functions-and-Spherical-Harmonics.html 0000664 0001750 0001750 00000012106 13055414563 022713 0 ustar edd eddNext: Logarithm and Related Functions, Previous: Lambert W Functions, Up: Special Functions [Index]
The Legendre Functions and Legendre Polynomials are described in Abramowitz & Stegun, Chapter 8. These functions are declared in the header file gsl_sf_legendre.h.
| • Legendre Polynomials: | ||
| • Associated Legendre Polynomials and Spherical Harmonics: | ||
| • Conical Functions: | ||
| • Radial Functions for Hyperbolic Space: |
Next: Histograms, Previous: Statistics, Up: Top [Index]
This chapter describes routines for computing running statistics, also known as online statistics, of data. These routines are suitable for handling large datasets for which it may be inconvenient or impractical to store in memory all at once. The data can be processed in a single pass, one point at a time. Each time a data point is added to the accumulator, internal parameters are updated in order to compute the current mean, variance, standard deviation, skewness, and kurtosis. These statistics are exact, and are updated with numerically stable single-pass algorithms. The median and arbitrary quantiles are also available, however these calculations use algorithms which provide approximations, and grow more accurate as more data is added to the accumulator.
The functions described in this chapter are declared in the header file gsl_rstat.h.
Next: Dilogarithm, Previous: Dawson Function, Up: Special Functions [Index]
The Debye functions D_n(x) are defined by the following integral,
D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1))
For further information see Abramowitz & Stegun, Section 27.1. The Debye functions are declared in the header file gsl_sf_debye.h.
These routines compute the first-order Debye function D_1(x) = (1/x) \int_0^x dt (t/(e^t - 1)).
These routines compute the second-order Debye function D_2(x) = (2/x^2) \int_0^x dt (t^2/(e^t - 1)).
These routines compute the third-order Debye function D_3(x) = (3/x^3) \int_0^x dt (t^3/(e^t - 1)).
These routines compute the fourth-order Debye function D_4(x) = (4/x^4) \int_0^x dt (t^4/(e^t - 1)).
These routines compute the fifth-order Debye function D_5(x) = (5/x^5) \int_0^x dt (t^5/(e^t - 1)).
These routines compute the sixth-order Debye function D_6(x) = (6/x^6) \int_0^x dt (t^6/(e^t - 1)).
Next: Sparse Matrices References and Further Reading, Previous: Sparse Matrices Conversion Between Sparse and Dense, Up: Sparse Matrices [Index]
The following example program builds a 5-by-4 sparse matrix and prints it in triplet, compressed column, and compressed row format. The matrix which is constructed is The output of the program is
printing all matrix elements: A(0,0) = 0 A(0,1) = 0 A(0,2) = 3.1 A(0,3) = 4.6 A(1,0) = 1 . . . A(4,0) = 4.1 A(4,1) = 0 A(4,2) = 0 A(4,3) = 0 matrix in triplet format (i,j,Aij): (0, 2, 3.1) (0, 3, 4.6) (1, 0, 1.0) (1, 2, 7.2) (3, 0, 2.1) (3, 1, 2.9) (3, 3, 8.5) (4, 0, 4.1) matrix in compressed column format: i = [ 1, 3, 4, 3, 0, 1, 0, 3, ] p = [ 0, 3, 4, 6, 8, ] d = [ 1, 2.1, 4.1, 2.9, 3.1, 7.2, 4.6, 8.5, ] matrix in compressed row format: i = [ 2, 3, 0, 2, 0, 1, 3, 0, ] p = [ 0, 2, 4, 4, 7, 8, ] d = [ 3.1, 4.6, 1, 7.2, 2.1, 2.9, 8.5, 4.1, ]
We see in the compressed column output, the data array stores each column contiguously, the array i stores the row index of the corresponding data element, and the array p stores the index of the start of each column in the data array. Similarly, for the compressed row output, the data array stores each row contiguously, the array i stores the column index of the corresponding data element, and the p array stores the index of the start of each row in the data array.
#include <stdio.h>
#include <stdlib.h>
#include <gsl/gsl_spmatrix.h>
int
main()
{
gsl_spmatrix *A = gsl_spmatrix_alloc(5, 4); /* triplet format */
gsl_spmatrix *B, *C;
size_t i, j;
/* build the sparse matrix */
gsl_spmatrix_set(A, 0, 2, 3.1);
gsl_spmatrix_set(A, 0, 3, 4.6);
gsl_spmatrix_set(A, 1, 0, 1.0);
gsl_spmatrix_set(A, 1, 2, 7.2);
gsl_spmatrix_set(A, 3, 0, 2.1);
gsl_spmatrix_set(A, 3, 1, 2.9);
gsl_spmatrix_set(A, 3, 3, 8.5);
gsl_spmatrix_set(A, 4, 0, 4.1);
printf("printing all matrix elements:\n");
for (i = 0; i < 5; ++i)
for (j = 0; j < 4; ++j)
printf("A(%zu,%zu) = %g\n", i, j,
gsl_spmatrix_get(A, i, j));
/* print out elements in triplet format */
printf("matrix in triplet format (i,j,Aij):\n");
gsl_spmatrix_fprintf(stdout, A, "%.1f");
/* convert to compressed column format */
B = gsl_spmatrix_ccs(A);
printf("matrix in compressed column format:\n");
printf("i = [ ");
for (i = 0; i < B->nz; ++i)
printf("%zu, ", B->i[i]);
printf("]\n");
printf("p = [ ");
for (i = 0; i < B->size2 + 1; ++i)
printf("%zu, ", B->p[i]);
printf("]\n");
printf("d = [ ");
for (i = 0; i < B->nz; ++i)
printf("%g, ", B->data[i]);
printf("]\n");
/* convert to compressed row format */
C = gsl_spmatrix_crs(A);
printf("matrix in compressed row format:\n");
printf("i = [ ");
for (i = 0; i < C->nz; ++i)
printf("%zu, ", C->i[i]);
printf("]\n");
printf("p = [ ");
for (i = 0; i < C->size1 + 1; ++i)
printf("%zu, ", C->p[i]);
printf("]\n");
printf("d = [ ");
for (i = 0; i < C->nz; ++i)
printf("%g, ", C->data[i]);
printf("]\n");
gsl_spmatrix_free(A);
gsl_spmatrix_free(B);
gsl_spmatrix_free(C);
return 0;
}
Next: Sparse Matrices References and Further Reading, Previous: Sparse Matrices Conversion Between Sparse and Dense, Up: Sparse Matrices [Index]
Previous: Sorting Examples, Up: Sorting [Index]
The subject of sorting is covered extensively in Knuth’s Sorting and Searching,
The Heapsort algorithm is described in the following book,
Next: Multiset functions, Previous: Accessing multiset elements, Up: Multisets [Index]
This function returns the range (n) of the multiset c.
This function returns the number of elements (k) in the multiset c.
This function returns a pointer to the array of elements in the multiset c.
This function checks that the multiset c is valid. The k elements should lie in the range 0 to n-1, with each value occurring in nondecreasing order.
Next: Reading and writing matrices, Previous: Accessing matrix elements, Up: Matrices [Index]
This function sets all the elements of the matrix m to the value x.
This function sets all the elements of the matrix m to zero.
This function sets the elements of the matrix m to the corresponding elements of the identity matrix, m(i,j) = \delta(i,j), i.e. a unit diagonal with all off-diagonal elements zero. This applies to both square and rectangular matrices.
Next: Sampling from a random number generator, Previous: The Random Number Generator Interface, Up: Random Number Generation [Index]
This function returns a pointer to a newly-created instance of a random number generator of type T. For example, the following code creates an instance of the Tausworthe generator,
gsl_rng * r = gsl_rng_alloc (gsl_rng_taus);
If there is insufficient memory to create the generator then the
function returns a null pointer and the error handler is invoked with an
error code of GSL_ENOMEM.
The generator is automatically initialized with the default seed,
gsl_rng_default_seed. This is zero by default but can be changed
either directly or by using the environment variable GSL_RNG_SEED
(see Random number environment variables).
The details of the available generator types are described later in this chapter.
This function initializes (or ‘seeds’) the random number generator. If
the generator is seeded with the same value of s on two different
runs, the same stream of random numbers will be generated by successive
calls to the routines below. If different values of s >= 1 are supplied, then the generated streams of random
numbers should be completely different. If the seed s is zero
then the standard seed from the original implementation is used
instead. For example, the original Fortran source code for the
ranlux generator used a seed of 314159265, and so choosing
s equal to zero reproduces this when using
gsl_rng_ranlux.
When using multiple seeds with the same generator, choose seed values greater than zero to avoid collisions with the default setting.
Note that the most generators only accept 32-bit seeds, with higher values being reduced modulo 2^32. For generators with smaller ranges the maximum seed value will typically be lower.
This function frees all the memory associated with the generator r.
Next: Sampling from a random number generator, Previous: The Random Number Generator Interface, Up: Random Number Generation [Index]
Previous: Level 3 CBLAS Functions, Up: GSL CBLAS Library [Index]
The following program computes the product of two matrices using the Level-3 BLAS function SGEMM,
[ 0.11 0.12 0.13 ] [ 1011 1012 ] [ 367.76 368.12 ]
[ 0.21 0.22 0.23 ] [ 1021 1022 ] = [ 674.06 674.72 ]
[ 1031 1032 ]
The matrices are stored in row major order but could be stored in column
major order if the first argument of the call to cblas_sgemm was
changed to CblasColMajor.
#include <stdio.h>
#include <gsl/gsl_cblas.h>
int
main (void)
{
int lda = 3;
float A[] = { 0.11, 0.12, 0.13,
0.21, 0.22, 0.23 };
int ldb = 2;
float B[] = { 1011, 1012,
1021, 1022,
1031, 1032 };
int ldc = 2;
float C[] = { 0.00, 0.00,
0.00, 0.00 };
/* Compute C = A B */
cblas_sgemm (CblasRowMajor,
CblasNoTrans, CblasNoTrans, 2, 2, 3,
1.0, A, lda, B, ldb, 0.0, C, ldc);
printf ("[ %g, %g\n", C[0], C[1]);
printf (" %g, %g ]\n", C[2], C[3]);
return 0;
}
To compile the program use the following command line,
$ gcc -Wall demo.c -lgslcblas
There is no need to link with the main library -lgsl in this
case as the CBLAS library is an independent unit. Here is the output
from the program,
$ ./a.out
[ 367.76, 368.12 674.06, 674.72 ]
Next: Multidimensional Root-Finding, Previous: One dimensional Root-Finding, Up: Top [Index]
This chapter describes routines for finding minima of arbitrary one-dimensional functions. The library provides low level components for a variety of iterative minimizers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the algorithms. Each class of methods uses the same framework, so that you can switch between minimizers at runtime without needing to recompile your program. Each instance of a minimizer keeps track of its own state, allowing the minimizers to be used in multi-threaded programs.
The header file gsl_min.h contains prototypes for the minimization functions and related declarations. To use the minimization algorithms to find the maximum of a function simply invert its sign.
Next: Further Information, Previous: No Warranty, Up: Introduction [Index]
A list of known bugs can be found in the BUGS file included in the GSL distribution or online in the GSL bug tracker.1 Details of compilation problems can be found in the INSTALL file.
If you find a bug which is not listed in these files, please report it to bug-gsl@gnu.org.
All bug reports should include:
It is useful if you can check whether the same problem occurs when the library is compiled without optimization. Thank you.
Any errors or omissions in this manual can also be reported to the same address.
gsl-ref-html-2.3/Arctangent-Integral.html 0000664 0001750 0001750 00000007731 13055414520 016461 0 ustar edd eddPrevious: Trigonometric Integrals, Up: Exponential Integrals [Index]
These routines compute the Arctangent integral, which is defined as AtanInt(x) = \int_0^x dt \arctan(t)/t.
Next: Copying vectors, Previous: Reading and writing vectors, Up: Vectors [Index]
In addition to creating vectors from slices of blocks it is also possible to slice vectors and create vector views. For example, a subvector of another vector can be described with a view, or two views can be made which provide access to the even and odd elements of a vector.
A vector view is a temporary object, stored on the stack, which can be
used to operate on a subset of vector elements. Vector views can be
defined for both constant and non-constant vectors, using separate types
that preserve constness. A vector view has the type
gsl_vector_view and a constant vector view has the type
gsl_vector_const_view. In both cases the elements of the view
can be accessed as a gsl_vector using the vector component
of the view object. A pointer to a vector of type gsl_vector *
or const gsl_vector * can be obtained by taking the address of
this component with the & operator.
When using this pointer it is important to ensure that the view itself
remains in scope—the simplest way to do so is by always writing the
pointer as &view.vector, and never storing this value
in another variable.
These functions return a vector view of a subvector of another vector v. The start of the new vector is offset by offset elements from the start of the original vector. The new vector has n elements. Mathematically, the i-th element of the new vector v’ is given by,
v'(i) = v->data[(offset + i)*v->stride]
where the index i runs from 0 to n-1.
The data pointer of the returned vector struct is set to null if
the combined parameters (offset,n) overrun the end of the
original vector.
The new vector is only a view of the block underlying the original vector, v. The block containing the elements of v is not owned by the new vector. When the view goes out of scope the original vector v and its block will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use.
The function gsl_vector_const_subvector is equivalent to
gsl_vector_subvector but can be used for vectors which are
declared const.
These functions return a vector view of a subvector of another vector
v with an additional stride argument. The subvector is formed in
the same way as for gsl_vector_subvector but the new vector has
n elements with a step-size of stride from one element to
the next in the original vector. Mathematically, the i-th element
of the new vector v’ is given by,
v'(i) = v->data[(offset + i*stride)*v->stride]
where the index i runs from 0 to n-1.
Note that subvector views give direct access to the underlying elements
of the original vector. For example, the following code will zero the
even elements of the vector v of length n, while leaving the
odd elements untouched,
gsl_vector_view v_even = gsl_vector_subvector_with_stride (v, 0, 2, n/2); gsl_vector_set_zero (&v_even.vector);
A vector view can be passed to any subroutine which takes a vector
argument just as a directly allocated vector would be, using
&view.vector. For example, the following code
computes the norm of the odd elements of v using the BLAS
routine DNRM2,
gsl_vector_view v_odd = gsl_vector_subvector_with_stride (v, 1, 2, n/2); double r = gsl_blas_dnrm2 (&v_odd.vector);
The function gsl_vector_const_subvector_with_stride is equivalent
to gsl_vector_subvector_with_stride but can be used for vectors
which are declared const.
These functions return a vector view of the real parts of the complex vector v.
The function gsl_vector_complex_const_real is equivalent to
gsl_vector_complex_real but can be used for vectors which are
declared const.
These functions return a vector view of the imaginary parts of the complex vector v.
The function gsl_vector_complex_const_imag is equivalent to
gsl_vector_complex_imag but can be used for vectors which are
declared const.
These functions return a vector view of an array. The start of the new vector is given by base and has n elements. Mathematically, the i-th element of the new vector v’ is given by,
v'(i) = base[i]
where the index i runs from 0 to n-1.
The array containing the elements of v is not owned by the new vector view. When the view goes out of scope the original array will continue to exist. The original memory can only be deallocated by freeing the original pointer base. Of course, the original array should not be deallocated while the view is still in use.
The function gsl_vector_const_view_array is equivalent to
gsl_vector_view_array but can be used for arrays which are
declared const.
These functions return a vector view of an array base with an
additional stride argument. The subvector is formed in the same way as
for gsl_vector_view_array but the new vector has n elements
with a step-size of stride from one element to the next in the
original array. Mathematically, the i-th element of the new
vector v’ is given by,
v'(i) = base[i*stride]
where the index i runs from 0 to n-1.
Note that the view gives direct access to the underlying elements of the
original array. A vector view can be passed to any subroutine which
takes a vector argument just as a directly allocated vector would be,
using &view.vector.
The function gsl_vector_const_view_array_with_stride is
equivalent to gsl_vector_view_array_with_stride but can be used
for arrays which are declared const.
Next: Copying vectors, Previous: Reading and writing vectors, Up: Vectors [Index]
Next: GCC warning options for numerical programs, Previous: Examining floating point registers, Up: Debugging Numerical Programs [Index]
It is possible to stop the program whenever a SIGFPE floating
point exception occurs. This can be useful for finding the cause of an
unexpected infinity or NaN. The current handler settings can be
shown with the command info signal SIGFPE.
(gdb) info signal SIGFPE Signal Stop Print Pass to program Description SIGFPE Yes Yes Yes Arithmetic exception
Unless the program uses a signal handler the default setting should be
changed so that SIGFPE is not passed to the program, as this would cause
it to exit. The command handle SIGFPE stop nopass prevents this.
(gdb) handle SIGFPE stop nopass Signal Stop Print Pass to program Description SIGFPE Yes Yes No Arithmetic exception
Depending on the platform it may be necessary to instruct the kernel to
generate signals for floating point exceptions. For programs using GSL
this can be achieved using the GSL_IEEE_MODE environment variable
in conjunction with the function gsl_ieee_env_setup as described
in see IEEE floating-point arithmetic.
(gdb) set env GSL_IEEE_MODE=double-precision
Next: Statistics, Previous: Quasi-Random Sequences, Up: Top [Index]
This chapter describes functions for generating random variates and computing their probability distributions. Samples from the distributions described in this chapter can be obtained using any of the random number generators in the library as an underlying source of randomness.
In the simplest cases a non-uniform distribution can be obtained analytically from the uniform distribution of a random number generator by applying an appropriate transformation. This method uses one call to the random number generator. More complicated distributions are created by the acceptance-rejection method, which compares the desired distribution against a distribution which is similar and known analytically. This usually requires several samples from the generator.
The library also provides cumulative distribution functions and inverse cumulative distribution functions, sometimes referred to as quantile functions. The cumulative distribution functions and their inverses are computed separately for the upper and lower tails of the distribution, allowing full accuracy to be retained for small results.
The functions for random variates and probability density functions described in this section are declared in gsl_randist.h. The corresponding cumulative distribution functions are declared in gsl_cdf.h.
Note that the discrete random variate functions always
return a value of type unsigned int, and on most platforms this
has a maximum value of 2^32-1 ~=~ 4.29e9. They should only be called with
a safe range of parameters (where there is a negligible probability of
a variate exceeding this limit) to prevent incorrect results due to
overflow.
Next: Statistics, Previous: Quasi-Random Sequences, Up: Top [Index]
Next: Physical Constant Examples, Previous: Force and Energy, Up: Physical Constants [Index]
These constants are dimensionless scaling factors.
GSL_CONST_NUM_YOTTA10^24
GSL_CONST_NUM_ZETTA10^21
GSL_CONST_NUM_EXA10^18
GSL_CONST_NUM_PETA10^15
GSL_CONST_NUM_TERA10^12
GSL_CONST_NUM_GIGA10^9
GSL_CONST_NUM_MEGA10^6
GSL_CONST_NUM_KILO10^3
GSL_CONST_NUM_MILLI10^-3
GSL_CONST_NUM_MICRO10^-6
GSL_CONST_NUM_NANO10^-9
GSL_CONST_NUM_PICO10^-12
GSL_CONST_NUM_FEMTO10^-15
GSL_CONST_NUM_ATTO10^-18
GSL_CONST_NUM_ZEPTO10^-21
GSL_CONST_NUM_YOCTO10^-24
Next: Multisets, Previous: Permutations, Up: Top [Index]
This chapter describes functions for creating and manipulating combinations. A combination c is represented by an array of k integers in the range 0 to n-1, where each value c_i occurs at most once. The combination c corresponds to indices of k elements chosen from an n element vector. Combinations are useful for iterating over all k-element subsets of a set.
The functions described in this chapter are defined in the header file gsl_combination.h.
Next: Linear Algebra, Previous: Sorting, Up: Top [Index]
The Basic Linear Algebra Subprograms (BLAS) define a set of fundamental operations on vectors and matrices which can be used to create optimized higher-level linear algebra functionality.
The library provides a low-level layer which corresponds directly to the C-language BLAS standard, referred to here as “CBLAS”, and a higher-level interface for operations on GSL vectors and matrices. Users who are interested in simple operations on GSL vector and matrix objects should use the high-level layer described in this chapter. The functions are declared in the file gsl_blas.h and should satisfy the needs of most users.
Note that GSL matrices are implemented using dense-storage so the interface only includes the corresponding dense-storage BLAS functions. The full BLAS functionality for band-format and packed-format matrices is available through the low-level CBLAS interface. Similarly, GSL vectors are restricted to positive strides, whereas the low-level CBLAS interface supports negative strides as specified in the BLAS standard.12
The interface for the gsl_cblas layer is specified in the file
gsl_cblas.h. This interface corresponds to the BLAS Technical
Forum’s standard for the C interface to legacy BLAS
implementations. Users who have access to other conforming CBLAS
implementations can use these in place of the version provided by the
library. Note that users who have only a Fortran BLAS library can
use a CBLAS conformant wrapper to convert it into a CBLAS
library. A reference CBLAS wrapper for legacy Fortran
implementations exists as part of the CBLAS standard and can
be obtained from Netlib. The complete set of CBLAS functions is
listed in an appendix (see GSL CBLAS Library).
There are three levels of BLAS operations,
Vector operations, e.g. y = \alpha x + y
Matrix-vector operations, e.g. y = \alpha A x + \beta y
Matrix-matrix operations, e.g. C = \alpha A B + C
Each routine has a name which specifies the operation, the type of matrices involved and their precisions. Some of the most common operations and their names are given below,
scalar product, x^T y
vector sum, \alpha x + y
matrix-vector product, A x
matrix-vector solve, inv(A) x
matrix-matrix product, A B
matrix-matrix solve, inv(A) B
The types of matrices are,
general
general band
symmetric
symmetric band
symmetric packed
hermitian
hermitian band
hermitian packed
triangular
triangular band
triangular packed
Each operation is defined for four precisions,
single real
double real
single complex
double complex
Thus, for example, the name SGEMM stands for “single-precision general matrix-matrix multiply” and ZGEMM stands for “double-precision complex matrix-matrix multiply”.
Note that the vector and matrix arguments to BLAS functions must not be aliased, as the results are undefined when the underlying arrays overlap (see Aliasing of arrays).
| • GSL BLAS Interface: | ||
| • BLAS Examples: | ||
| • BLAS References and Further Reading: |
In the low-level CBLAS interface, a negative stride accesses the vector elements in reverse order, i.e. the i-th element is given by (N-i)*|incx| for incx < 0.
Next: Linear Algebra, Previous: Sorting, Up: Top [Index]
Previous: Level 2 GSL BLAS Interface, Up: GSL BLAS Interface [Index]
These functions compute the matrix-matrix product and sum C =
\alpha op(A) op(B) + \beta C where op(A) = A, A^T,
A^H for TransA = CblasNoTrans, CblasTrans,
CblasConjTrans and similarly for the parameter TransB.
These functions compute the matrix-matrix product and sum C =
\alpha A B + \beta C for Side is CblasLeft and C =
\alpha B A + \beta C for Side is CblasRight, where the
matrix A is symmetric. When Uplo is CblasUpper then
the upper triangle and diagonal of A are used, and when Uplo
is CblasLower then the lower triangle and diagonal of A are
used.
These functions compute the matrix-matrix product and sum C =
\alpha A B + \beta C for Side is CblasLeft and C =
\alpha B A + \beta C for Side is CblasRight, where the
matrix A is hermitian. When Uplo is CblasUpper then
the upper triangle and diagonal of A are used, and when Uplo
is CblasLower then the lower triangle and diagonal of A are
used. The imaginary elements of the diagonal are automatically set to
zero.
These functions compute the matrix-matrix product B = \alpha op(A)
B for Side is CblasLeft and B = \alpha B op(A) for
Side is CblasRight. The matrix A is triangular and
op(A) = A, A^T, A^H for TransA =
CblasNoTrans, CblasTrans, CblasConjTrans. When
Uplo is CblasUpper then the upper triangle of A is
used, and when Uplo is CblasLower then the lower triangle
of A is used. If Diag is CblasNonUnit then the
diagonal of A is used, but if Diag is CblasUnit then
the diagonal elements of the matrix A are taken as unity and are
not referenced.
These functions compute the inverse-matrix matrix product
B = \alpha op(inv(A))B for Side is
CblasLeft and B = \alpha B op(inv(A)) for
Side is CblasRight. The matrix A is triangular and
op(A) = A, A^T, A^H for TransA =
CblasNoTrans, CblasTrans, CblasConjTrans. When
Uplo is CblasUpper then the upper triangle of A is
used, and when Uplo is CblasLower then the lower triangle
of A is used. If Diag is CblasNonUnit then the
diagonal of A is used, but if Diag is CblasUnit then
the diagonal elements of the matrix A are taken as unity and are
not referenced.
These functions compute a rank-k update of the symmetric matrix C,
C = \alpha A A^T + \beta C when Trans is
CblasNoTrans and C = \alpha A^T A + \beta C when
Trans is CblasTrans. Since the matrix C is symmetric
only its upper half or lower half need to be stored. When Uplo is
CblasUpper then the upper triangle and diagonal of C are
used, and when Uplo is CblasLower then the lower triangle
and diagonal of C are used.
These functions compute a rank-k update of the hermitian matrix C,
C = \alpha A A^H + \beta C when Trans is
CblasNoTrans and C = \alpha A^H A + \beta C when
Trans is CblasConjTrans. Since the matrix C is hermitian
only its upper half or lower half need to be stored. When Uplo is
CblasUpper then the upper triangle and diagonal of C are
used, and when Uplo is CblasLower then the lower triangle
and diagonal of C are used. The imaginary elements of the
diagonal are automatically set to zero.
These functions compute a rank-2k update of the symmetric matrix C,
C = \alpha A B^T + \alpha B A^T + \beta C when Trans is
CblasNoTrans and C = \alpha A^T B + \alpha B^T A + \beta C when
Trans is CblasTrans. Since the matrix C is symmetric
only its upper half or lower half need to be stored. When Uplo is
CblasUpper then the upper triangle and diagonal of C are
used, and when Uplo is CblasLower then the lower triangle
and diagonal of C are used.
These functions compute a rank-2k update of the hermitian matrix C,
C = \alpha A B^H + \alpha^* B A^H + \beta C when Trans is
CblasNoTrans and C = \alpha A^H B + \alpha^* B^H A + \beta C when
Trans is CblasConjTrans. Since the matrix C is hermitian
only its upper half or lower half need to be stored. When Uplo is
CblasUpper then the upper triangle and diagonal of C are
used, and when Uplo is CblasLower then the lower triangle
and diagonal of C are used. The imaginary elements of the
diagonal are automatically set to zero.
Previous: Level 2 GSL BLAS Interface, Up: GSL BLAS Interface [Index]
Next: Nonlinear Least-Squares Fitting, Previous: Multidimensional Minimization, Up: Top [Index]
This chapter describes routines for performing least squares fits to experimental data using linear combinations of functions. The data may be weighted or unweighted, i.e. with known or unknown errors. For weighted data the functions compute the best fit parameters and their associated covariance matrix. For unweighted data the covariance matrix is estimated from the scatter of the points, giving a variance-covariance matrix.
The functions are divided into separate versions for simple one- or two-parameter regression and multiple-parameter fits.
Next: The Logistic Distribution, Previous: The t-distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the beta distribution. The distribution function is,
p(x) dx = {\Gamma(a+b) \over \Gamma(a) \Gamma(b)} x^{a-1} (1-x)^{b-1} dx
for 0 <= x <= 1.
This function computes the probability density p(x) at x for a beta distribution with parameters a and b, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the beta distribution with parameters a and b.
Next: Updating and accessing 2D histogram elements, Previous: 2D Histogram allocation, Up: Histograms [Index]
This function copies the histogram src into the pre-existing histogram dest, making dest into an exact copy of src. The two histograms must be of the same size.
This function returns a pointer to a newly created histogram which is an exact copy of the histogram src.
Next: Coupling Coefficients, Previous: Clausen Functions, Up: Special Functions [Index]
The prototypes of the Coulomb functions are declared in the header file gsl_sf_coulomb.h. Both bound state and scattering solutions are available.
| • Normalized Hydrogenic Bound States: | ||
| • Coulomb Wave Functions: | ||
| • Coulomb Wave Function Normalization Constant: |
Next: Troubleshooting, Previous: Robust linear regression, Up: Least-Squares Fitting [Index]
This module is concerned with solving large dense least squares systems X c = y where the n-by-p matrix X has n >> p (ie: many more rows than columns). This type of matrix is called a “tall skinny” matrix, and for some applications, it may not be possible to fit the entire matrix in memory at once to use the standard SVD approach. Therefore, the algorithms in this module are designed to allow the user to construct smaller blocks of the matrix X and accumulate those blocks into the larger system one at a time. The algorithms in this module never need to store the entire matrix X in memory. The large linear least squares routines support data weights and Tikhonov regularization, and are designed to minimize the residual
\chi^2 = || y - Xc ||_W^2 + \lambda^2 || L c ||^2
where y is the n-by-1 observation vector, X is the n-by-p design matrix, c is the p-by-1 solution vector, W = diag(w_1,...,w_n) is the data weighting matrix, L is an m-by-p regularization matrix, \lambda is a regularization parameter, and ||r||_W^2 = r^T W r. In the discussion which follows, we will assume that the system has been converted into Tikhonov standard form,
\chi^2 = || y~ - X~ c~ ||^2 + \lambda^2 || c~ ||^2
and we will drop the tilde characters from the various parameters. For a discussion of the transformation to standard form see Regularized regression.
The basic idea is to partition the matrix X and observation vector y as
[ X_1 ] c = [ y_1 ] [ X_2 ] [ y_2 ] [ X_3 ] [ y_3 ] [ ... ] [ ... ] [ X_k ] [ y_k ]
into k blocks, where each block (X_i,y_i) may have any number of rows, but each X_i has p columns. The sections below describe the methods available for solving this partitioned system. The functions are declared in the header file gsl_multilarge.h.
| • Large Dense Linear Systems Normal Equations: | ||
| • Large Dense Linear Systems TSQR: | ||
| • Large Dense Linear Systems Solution Steps: | ||
| • Large Dense Linear Systems Routines: |
Next: Troubleshooting, Previous: Robust linear regression, Up: Least-Squares Fitting [Index]
Next: Printers Units, Previous: Imperial Units, Up: Physical Constants [Index]
GSL_CONST_MKSA_KILOMETERS_PER_HOURThe speed of 1 kilometer per hour.
GSL_CONST_MKSA_MILES_PER_HOURThe speed of 1 mile per hour.
GSL_CONST_MKSA_NAUTICAL_MILEThe length of 1 nautical mile.
GSL_CONST_MKSA_FATHOMThe length of 1 fathom.
GSL_CONST_MKSA_KNOTThe speed of 1 knot.
Next: The Levy skew alpha-Stable Distribution, Previous: The Landau Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the Levy symmetric stable distribution with scale c and exponent alpha. The symmetric stable probability distribution is defined by a Fourier transform,
p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha)
There is no explicit solution for the form of p(x) and the
library does not define a corresponding pdf function. For
\alpha = 1 the distribution reduces to the Cauchy distribution. For
\alpha = 2 it is a Gaussian distribution with \sigma = \sqrt{2} c. For \alpha < 1 the tails of the
distribution become extremely wide.
The algorithm only works for 0 < alpha <= 2.
Next: Inverse Complex Trigonometric Functions, Previous: Elementary Complex Functions, Up: Complex Numbers [Index]
This function returns the complex sine of the complex number z, \sin(z) = (\exp(iz) - \exp(-iz))/(2i).
This function returns the complex cosine of the complex number z, \cos(z) = (\exp(iz) + \exp(-iz))/2.
This function returns the complex tangent of the complex number z, \tan(z) = \sin(z)/\cos(z).
This function returns the complex secant of the complex number z, \sec(z) = 1/\cos(z).
This function returns the complex cosecant of the complex number z, \csc(z) = 1/\sin(z).
This function returns the complex cotangent of the complex number z, \cot(z) = 1/\tan(z).
Previous: Conical Functions, Up: Legendre Functions and Spherical Harmonics [Index]
The following spherical functions are specializations of Legendre functions which give the regular eigenfunctions of the Laplacian on a 3-dimensional hyperbolic space H3d. Of particular interest is the flat limit, \lambda \to \infty, \eta \to 0, \lambda\eta fixed.
These routines compute the zeroth radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, L^{H3d}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta)) for \eta >= 0. In the flat limit this takes the form L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta).
These routines compute the first radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, L^{H3d}_1(\lambda,\eta) := 1/\sqrt{\lambda^2 + 1} \sin(\lambda \eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda \cot(\lambda\eta)) for \eta >= 0. In the flat limit this takes the form L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta).
These routines compute the l-th radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space \eta >= 0, l >= 0. In the flat limit this takes the form L^{H3d}_l(\lambda,\eta) = j_l(\lambda\eta).
This function computes an array of radial eigenfunctions L^{H3d}_l(\lambda, \eta) for 0 <= l <= lmax.
Next: Minimization References and Further Reading, Previous: Minimization Algorithms, Up: One dimensional Minimization [Index]
The following program uses the Brent algorithm to find the minimum of the function f(x) = \cos(x) + 1, which occurs at x = \pi. The starting interval is (0,6), with an initial guess for the minimum of 2.
#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_min.h>
double fn1 (double x, void * params)
{
(void)(params); /* avoid unused parameter warning */
return cos(x) + 1.0;
}
int
main (void)
{
int status;
int iter = 0, max_iter = 100;
const gsl_min_fminimizer_type *T;
gsl_min_fminimizer *s;
double m = 2.0, m_expected = M_PI;
double a = 0.0, b = 6.0;
gsl_function F;
F.function = &fn1;
F.params = 0;
T = gsl_min_fminimizer_brent;
s = gsl_min_fminimizer_alloc (T);
gsl_min_fminimizer_set (s, &F, m, a, b);
printf ("using %s method\n",
gsl_min_fminimizer_name (s));
printf ("%5s [%9s, %9s] %9s %10s %9s\n",
"iter", "lower", "upper", "min",
"err", "err(est)");
printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
iter, a, b,
m, m - m_expected, b - a);
do
{
iter++;
status = gsl_min_fminimizer_iterate (s);
m = gsl_min_fminimizer_x_minimum (s);
a = gsl_min_fminimizer_x_lower (s);
b = gsl_min_fminimizer_x_upper (s);
status
= gsl_min_test_interval (a, b, 0.001, 0.0);
if (status == GSL_SUCCESS)
printf ("Converged:\n");
printf ("%5d [%.7f, %.7f] "
"%.7f %+.7f %.7f\n",
iter, a, b,
m, m - m_expected, b - a);
}
while (status == GSL_CONTINUE && iter < max_iter);
gsl_min_fminimizer_free (s);
return status;
}
Here are the results of the minimization procedure.
$ ./a.out
using brent method
iter [ lower, upper] min err err(est)
0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
1 [2.0000000, 6.0000000] 3.5278640 +0.3862713 4.0000000
2 [2.0000000, 3.5278640] 3.1748217 +0.0332290 1.5278640
3 [2.0000000, 3.1748217] 3.1264576 -0.0151351 1.1748217
4 [3.1264576, 3.1748217] 3.1414743 -0.0001183 0.0483641
5 [3.1414743, 3.1748217] 3.1415930 +0.0000004 0.0333474
Converged:
6 [3.1414743, 3.1415930] 3.1415927 +0.0000000 0.0001187
Next: The Lognormal Distribution, Previous: The Gamma Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the flat (uniform) distribution from a to b. The distribution is,
p(x) dx = {1 \over (b-a)} dx
if a <= x < b and 0 otherwise.
This function computes the probability density p(x) at x for a uniform distribution from a to b, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for a uniform distribution from a to b.
Next: Mathieu Functions, Previous: Legendre Functions and Spherical Harmonics, Up: Special Functions [Index]
Information on the properties of the Logarithm function can be found in Abramowitz & Stegun, Chapter 4. The functions described in this section are declared in the header file gsl_sf_log.h.
These routines compute the logarithm of x, \log(x), for x > 0.
These routines compute the logarithm of the magnitude of x, \log(|x|), for x \ne 0.
This routine computes the complex logarithm of z = z_r + i z_i. The results are returned as lnr, theta such that \exp(lnr + i \theta) = z_r + i z_i, where \theta lies in the range [-\pi,\pi].
These routines compute \log(1 + x) for x > -1 using an algorithm that is accurate for small x.
These routines compute \log(1 + x) - x for x > -1 using an algorithm that is accurate for small x.
Next: Auxiliary quasi-random number generator functions, Previous: Quasi-random number generator initialization, Up: Quasi-Random Sequences [Index]
This function stores the next point from the sequence generator q
in the array x. The space available for x must match the
dimension of the generator. The point x will lie in the range
0 < x_i < 1 for each x_i. An inline version of this function is used when HAVE_INLINE is defined.
Next: Hessenberg Decomposition of Real Matrices, Previous: Tridiagonal Decomposition of Real Symmetric Matrices, Up: Linear Algebra [Index]
A hermitian matrix A can be factorized by similarity transformations into the form,
A = U T U^T
where U is a unitary matrix and T is a real symmetric tridiagonal matrix.
This function factorizes the hermitian matrix A into the symmetric tridiagonal decomposition U T U^T. On output the real parts of the diagonal and subdiagonal part of the input matrix A contain the tridiagonal matrix T. The remaining lower triangular part of the input matrix contains the Householder vectors which, together with the Householder coefficients tau, encode the unitary matrix U. This storage scheme is the same as used by LAPACK. The upper triangular part of A and imaginary parts of the diagonal are not referenced.
This function unpacks the encoded tridiagonal decomposition (A,
tau) obtained from gsl_linalg_hermtd_decomp into the
unitary matrix U, the real vector of diagonal elements diag and
the real vector of subdiagonal elements subdiag.
This function unpacks the diagonal and subdiagonal of the encoded
tridiagonal decomposition (A, tau) obtained from the
gsl_linalg_hermtd_decomp into the real vectors
diag and subdiag.
Next: Discrete Hankel Transforms, Previous: Series Acceleration, Up: Top [Index]
This chapter describes functions for performing Discrete Wavelet Transforms (DWTs). The library includes wavelets for real data in both one and two dimensions. The wavelet functions are declared in the header files gsl_wavelet.h and gsl_wavelet2d.h.
| • DWT Definitions: | ||
| • DWT Initialization: | ||
| • DWT Transform Functions: | ||
| • DWT Examples: | ||
| • DWT References: |
Next: CQUAD doubly-adaptive integration, Previous: QAWO adaptive integration for oscillatory functions, Up: Numerical Integration [Index]
This function attempts to compute a Fourier integral of the function f over the semi-infinite interval [a,+\infty).
I = \int_a^{+\infty} dx f(x) sin(omega x)
I = \int_a^{+\infty} dx f(x) cos(omega x)
The parameter \omega and choice of \sin or \cos is taken from the table wf (the length L can take any value, since it is overridden by this function to a value appropriate for the Fourier integration). The integral is computed using the QAWO algorithm over each of the subintervals,
C_1 = [a, a + c] C_2 = [a + c, a + 2 c] ... = ... C_k = [a + (k-1) c, a + k c]
where c = (2 floor(|\omega|) + 1) \pi/|\omega|. The width c is chosen to cover an odd number of periods so that the contributions from the intervals alternate in sign and are monotonically decreasing when f is positive and monotonically decreasing. The sum of this sequence of contributions is accelerated using the epsilon-algorithm.
This function works to an overall absolute tolerance of abserr. The following strategy is used: on each interval C_k the algorithm tries to achieve the tolerance
TOL_k = u_k abserr
where u_k = (1 - p)p^{k-1} and p = 9/10. The sum of the geometric series of contributions from each interval gives an overall tolerance of abserr.
If the integration of a subinterval leads to difficulties then the accuracy requirement for subsequent intervals is relaxed,
TOL_k = u_k max(abserr, max_{i<k}{E_i})
where E_k is the estimated error on the interval C_k.
The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals is given by limit, which may not exceed the allocated size of the workspace. The integration over each subinterval uses the memory provided by cycle_workspace as workspace for the QAWO algorithm.
Next: CQUAD doubly-adaptive integration, Previous: QAWO adaptive integration for oscillatory functions, Up: Numerical Integration [Index]
Next: Multimin Caveats, Up: Multidimensional Minimization [Index]
The problem of multidimensional minimization requires finding a point x such that the scalar function,
f(x_1, …, x_n)
takes a value which is lower than at any neighboring point. For smooth functions the gradient g = \nabla f vanishes at the minimum. In general there are no bracketing methods available for the minimization of n-dimensional functions. The algorithms proceed from an initial guess using a search algorithm which attempts to move in a downhill direction.
Algorithms making use of the gradient of the function perform a one-dimensional line minimisation along this direction until the lowest point is found to a suitable tolerance. The search direction is then updated with local information from the function and its derivatives, and the whole process repeated until the true n-dimensional minimum is found.
Algorithms which do not require the gradient of the function use different strategies. For example, the Nelder-Mead Simplex algorithm maintains n+1 trial parameter vectors as the vertices of a n-dimensional simplex. On each iteration it tries to improve the worst vertex of the simplex by geometrical transformations. The iterations are continued until the overall size of the simplex has decreased sufficiently.
Both types of algorithms use a standard framework. The user provides a high-level driver for the algorithms, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,
Each iteration step consists either of an improvement to the
line-minimisation in the current direction or an update to the search
direction itself. The state for the minimizers is held in a
gsl_multimin_fdfminimizer struct or a
gsl_multimin_fminimizer struct.
Next: Multimin Caveats, Up: Multidimensional Minimization [Index]
Previous: Reading and writing blocks, Up: Blocks [Index]
The following program shows how to allocate a block,
#include <stdio.h>
#include <gsl/gsl_block.h>
int
main (void)
{
gsl_block * b = gsl_block_alloc (100);
printf ("length of block = %zu\n", b->size);
printf ("block data address = %p\n", b->data);
gsl_block_free (b);
return 0;
}
Here is the output from the program,
length of block = 100 block data address = 0x804b0d8
Next: Fixed order Gauss-Legendre integration, Previous: QAWF adaptive integration for Fourier integrals, Up: Numerical Integration [Index]
CQUAD is a new doubly-adaptive general-purpose quadrature
routine which can handle most types of singularities,
non-numerical function values such as Inf or NaN,
as well as some divergent integrals. It generally requires more
function evaluations than the integration routines in
QUADPACK, yet fails less often for difficult integrands.
The underlying algorithm uses a doubly-adaptive scheme in which Clenshaw-Curtis quadrature rules of increasing degree are used to compute the integral in each interval. The L_2-norm of the difference between the underlying interpolatory polynomials of two successive rules is used as an error estimate. The interval is subdivided if the difference between two successive rules is too large or a rule of maximum degree has been reached.
This function allocates a workspace sufficient to hold the data for n intervals. The number n is not the maximum number of intervals that will be evaluated. If the workspace is full, intervals with smaller error estimates will be discarded. A minimum of 3 intervals is required and for most functions, a workspace of size 100 is sufficient.
This function frees the memory associated with the workspace w.
This function computes the integral of f over (a,b) within the desired absolute and relative error limits, epsabs and epsrel using the CQUAD algorithm. The function returns the final approximation, result, an estimate of the absolute error, abserr, and the number of function evaluations required, nevals.
The CQUAD algorithm divides the integration region into subintervals, and in each iteration, the subinterval with the largest estimated error is processed. The algorithm uses Clenshaw-Curits quadrature rules of degree 4, 8, 16 and 32 over 5, 9, 17 and 33 nodes respectively. Each interval is initialized with the lowest-degree rule. When an interval is processed, the next-higher degree rule is evaluated and an error estimate is computed based on the L_2-norm of the difference between the underlying interpolating polynomials of both rules. If the highest-degree rule has already been used, or the interpolatory polynomials differ significantly, the interval is bisected.
The subintervals and their results are stored in the memory
provided by workspace. If the error estimate or the number of
function evaluations is not needed, the pointers abserr and nevals
can be set to NULL.
Next: Fixed order Gauss-Legendre integration, Previous: QAWF adaptive integration for Fourier integrals, Up: Numerical Integration [Index]
Next: 2D Introduction to Interpolation, Previous: 1D Interpolation Example programs, Up: Interpolation [Index]
Descriptions of the interpolation algorithms and further references can be found in the following publications:
Next: Using GSL error reporting in your own functions, Previous: Error Codes, Up: Error Handling [Index]
The default behavior of the GSL error handler is to print a short
message and call abort. When this default is in use programs
will stop with a core-dump whenever a library routine reports an error.
This is intended as a fail-safe default for programs which do not check
the return status of library routines (we don’t encourage you to write
programs this way).
If you turn off the default error handler it is your responsibility to check the return values of routines and handle them yourself. You can also customize the error behavior by providing a new error handler. For example, an alternative error handler could log all errors to a file, ignore certain error conditions (such as underflows), or start the debugger and attach it to the current process when an error occurs.
All GSL error handlers have the type gsl_error_handler_t, which is
defined in gsl_errno.h,
This is the type of GSL error handler functions. An error handler will
be passed four arguments which specify the reason for the error (a
string), the name of the source file in which it occurred (also a
string), the line number in that file (an integer) and the error number
(an integer). The source file and line number are set at compile time
using the __FILE__ and __LINE__ directives in the
preprocessor. An error handler function returns type void.
Error handler functions should be defined like this,
void handler (const char * reason,
const char * file,
int line,
int gsl_errno)
To request the use of your own error handler you need to call the
function gsl_set_error_handler which is also declared in
gsl_errno.h,
This function sets a new error handler, new_handler, for the GSL library routines. The previous handler is returned (so that you can restore it later). Note that the pointer to a user defined error handler function is stored in a static variable, so there can be only one error handler per program. This function should be not be used in multi-threaded programs except to set up a program-wide error handler from a master thread. The following example shows how to set and restore a new error handler,
/* save original handler, install new handler */ old_handler = gsl_set_error_handler (&my_handler); /* code uses new handler */ ..... /* restore original handler */ gsl_set_error_handler (old_handler);
To use the default behavior (abort on error) set the error
handler to NULL,
old_handler = gsl_set_error_handler (NULL);
This function turns off the error handler by defining an error handler which does nothing. This will cause the program to continue after any error, so the return values from any library routines must be checked. This is the recommended behavior for production programs. The previous handler is returned (so that you can restore it later).
The error behavior can be changed for specific applications by
recompiling the library with a customized definition of the
GSL_ERROR macro in the file gsl_errno.h.
Next: Using GSL error reporting in your own functions, Previous: Error Codes, Up: Error Handling [Index]
Previous: DWT Examples, Up: Wavelet Transforms [Index]
The mathematical background to wavelet transforms is covered in the original lectures by Daubechies,
An easy to read introduction to the subject with an emphasis on the application of the wavelet transform in various branches of science is,
For extensive coverage of signal analysis by wavelets, wavelet packets and local cosine bases see,
The concept of multiresolution analysis underlying the wavelet transform is described in,
The coefficients for the individual wavelet families implemented by the library can be found in the following papers,
The PhysioNet archive of physiological datasets can be found online at http://www.physionet.org/ and is described in the following paper,
Previous: DWT Examples, Up: Wavelet Transforms [Index]
Next: Alternative optimized functions, Previous: Long double, Up: Using the library [Index]
To help in writing portable applications GSL provides some implementations of functions that are found in other libraries, such as the BSD math library. You can write your application to use the native versions of these functions, and substitute the GSL versions via a preprocessor macro if they are unavailable on another platform.
For example, after determining whether the BSD function hypot is
available you can include the following macro definitions in a file
config.h with your application,
/* Substitute gsl_hypot for missing system hypot */ #ifndef HAVE_HYPOT #define hypot gsl_hypot #endif
The application source files can then use the include command
#include <config.h> to replace each occurrence of hypot by
gsl_hypot when hypot is not available. This substitution
can be made automatically if you use autoconf, see Autoconf Macros.
In most circumstances the best strategy is to use the native versions of these functions when available, and fall back to GSL versions otherwise, since this allows your application to take advantage of any platform-specific optimizations in the system library. This is the strategy used within GSL itself.
gsl-ref-html-2.3/Histograms.html 0000664 0001750 0001750 00000023165 13055414422 014750 0 ustar edd eddNext: N-tuples, Previous: Running Statistics, Up: Top [Index]
This chapter describes functions for creating histograms. Histograms provide a convenient way of summarizing the distribution of a set of data. A histogram consists of a set of bins which count the number of events falling into a given range of a continuous variable x. In GSL the bins of a histogram contain floating-point numbers, so they can be used to record both integer and non-integer distributions. The bins can use arbitrary sets of ranges (uniformly spaced bins are the default). Both one and two-dimensional histograms are supported.
Once a histogram has been created it can also be converted into a probability distribution function. The library provides efficient routines for selecting random samples from probability distributions. This can be useful for generating simulations based on real data.
The functions are declared in the header files gsl_histogram.h and gsl_histogram2d.h.
Next: N-tuples, Previous: Running Statistics, Up: Top [Index]
Next: Pochhammer Symbol, Previous: Gamma Functions, Up: Gamma and Beta Functions [Index]
Although factorials can be computed from the Gamma function, using the relation n! = \Gamma(n+1) for non-negative integer n, it is usually more efficient to call the functions in this section, particularly for small values of n, whose factorial values are maintained in hardcoded tables.
These routines compute the factorial n!. The factorial is
related to the Gamma function by n! = \Gamma(n+1).
The maximum value of n such that n! is not
considered an overflow is given by the macro GSL_SF_FACT_NMAX
and is 170.
These routines compute the double factorial n!! = n(n-2)(n-4) \dots.
The maximum value of n such that n!! is not
considered an overflow is given by the macro GSL_SF_DOUBLEFACT_NMAX
and is 297.
These routines compute the logarithm of the factorial of n,
\log(n!). The algorithm is faster than computing
\ln(\Gamma(n+1)) via gsl_sf_lngamma for n < 170,
but defers for larger n.
These routines compute the logarithm of the double factorial of n, \log(n!!).
These routines compute the combinatorial factor n choose m
= n!/(m!(n-m)!)
These routines compute the logarithm of n choose m. This is
equivalent to the sum \log(n!) - \log(m!) - \log((n-m)!).
These routines compute the Taylor coefficient x^n / n! for x >= 0, n >= 0.
Next: Pochhammer Symbol, Previous: Gamma Functions, Up: Gamma and Beta Functions [Index]
Next: Tridiagonal Decomposition of Real Symmetric Matrices, Previous: Pivoted Cholesky Decomposition, Up: Linear Algebra [Index]
The modified Cholesky decomposition is suitable for solving systems A x = b where A is a symmetric indefinite matrix. Such matrices arise in nonlinear optimization algorithms. The standard Cholesky decomposition requires a positive definite matrix and would fail in this case. Instead of resorting to a method like QR or SVD, which do not take into account the symmetry of the matrix, we can instead introduce a small perturbation to the matrix A to make it positive definite, and then use a Cholesky decomposition on the perturbed matrix. The resulting decomposition satisfies
P (A + E) P^T = L D L^T
where P is a permutation matrix, E is a diagonal perturbation matrix, L is unit lower triangular, and D is diagonal. If A is sufficiently positive definite, then the perturbation matrix E will be zero and this method is equivalent to the pivoted Cholesky algorithm. For indefinite matrices, the perturbation matrix E is computed to ensure that A + E is positive definite and well conditioned.
This function factors the symmetric, indefinite square matrix A into the Modified Cholesky decomposition P (A + E) P^T = L D L^T. On input, the values from the diagonal and lower-triangular part of the matrix A are used to construct the factorization. On output the diagonal of the input matrix A stores the diagonal elements of D, and the lower triangular portion of A contains the matrix L. Since L has ones on its diagonal these do not need to be explicitely stored. The upper triangular portion of A is unmodified. The permutation matrix P is stored in p on output. The diagonal perturbation matrix is stored in E on output. The parameter E may be set to NULL if it is not required.
This function solves the perturbed system (A + E) x = b using the Cholesky
decomposition of A + E held in the matrix LDLT and permutation
p which must have been previously computed by gsl_linalg_mcholesky_decomp.
This function solves the perturbed system (A + E) x = b in-place using the Cholesky
decomposition of A + E held in the matrix LDLT and permutation
p which must have been previously computed by gsl_linalg_mcholesky_decomp.
On input, x contains the right hand side vector b which is
replaced by the solution vector on output.
This function estimates the reciprocal condition number (using the 1-norm) of the perturbed matrix A + E, using its pivoted Cholesky decomposition provided in LDLT. The reciprocal condition number estimate, defined as 1 / (||A + E||_1 \cdot ||(A + E)^{-1}||_1), is stored in rcond. Additional workspace of size 3 N is required in work.
Next: Tridiagonal Decomposition of Real Symmetric Matrices, Previous: Pivoted Cholesky Decomposition, Up: Linear Algebra [Index]
Next: QAGI adaptive integration on infinite intervals, Previous: QAGS adaptive integration with singularities, Up: Numerical Integration [Index]
This function applies the adaptive integration algorithm QAGS taking account of the user-supplied locations of singular points. The array pts of length npts should contain the endpoints of the integration ranges defined by the integration region and locations of the singularities. For example, to integrate over the region (a,b) with break-points at x_1, x_2, x_3 (where a < x_1 < x_2 < x_3 < b) the following pts array should be used
pts[0] = a pts[1] = x_1 pts[2] = x_2 pts[3] = x_3 pts[4] = b
with npts = 5.
If you know the locations of the singular points in the integration
region then this routine will be faster than QAGS.
Next: Reading ntuples, Previous: Opening an existing ntuple file, Up: N-tuples [Index]
This function writes the current ntuple ntuple->ntuple_data of size ntuple->size to the corresponding file.
This function is a synonym for gsl_ntuple_write.
Next: Histogram Operations, Previous: Searching histogram ranges, Up: Histograms [Index]
This function returns the maximum value contained in the histogram bins.
This function returns the index of the bin containing the maximum value. In the case where several bins contain the same maximum value the smallest index is returned.
This function returns the minimum value contained in the histogram bins.
This function returns the index of the bin containing the minimum value. In the case where several bins contain the same maximum value the smallest index is returned.
This function returns the mean of the histogrammed variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation. The accuracy of the result is limited by the bin width.
This function returns the standard deviation of the histogrammed variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation. The accuracy of the result is limited by the bin width.
This function returns the sum of all bin values. Negative bin values are included in the sum.
Next: Vectors, Previous: Data types, Up: Vectors and Matrices [Index]
For consistency all memory is allocated through a gsl_block
structure. The structure contains two components, the size of an area of
memory and a pointer to the memory. The gsl_block structure looks
like this,
typedef struct
{
size_t size;
double * data;
} gsl_block;
Vectors and matrices are made by slicing an underlying block. A slice is a set of elements formed from an initial offset and a combination of indices and step-sizes. In the case of a matrix the step-size for the column index represents the row-length. The step-size for a vector is known as the stride.
The functions for allocating and deallocating blocks are defined in gsl_block.h
| • Block allocation: | ||
| • Reading and writing blocks: | ||
| • Example programs for blocks: |
Next: Multimin Iteration, Previous: Initializing the Multidimensional Minimizer, Up: Multidimensional Minimization [Index]
You must provide a parametric function of n variables for the minimizers to operate on. You may also need to provide a routine which calculates the gradient of the function and a third routine which calculates both the function value and the gradient together. In order to allow for general parameters the functions are defined by the following data types:
This data type defines a general function of n variables with parameters and the corresponding gradient vector of derivatives,
double (* f) (const gsl_vector * x, void * params)this function should return the result
f(x,params) for argument x and parameters params.
If the function cannot be computed, an error value of GSL_NAN
should be returned.
void (* df) (const gsl_vector * x, void * params, gsl_vector * g)this function should store the n-dimensional gradient g_i = d f(x,params) / d x_i in the vector g for argument x and parameters params, returning an appropriate error code if the function cannot be computed.
void (* fdf) (const gsl_vector * x, void * params, double * f, gsl_vector * g)This function should set the values of the f and g as above, for arguments x and parameters params. This function provides an optimization of the separate functions for f(x) and g(x)—it is always faster to compute the function and its derivative at the same time.
size_t nthe dimension of the system, i.e. the number of components of the vectors x.
void * paramsa pointer to the parameters of the function.
This data type defines a general function of n variables with parameters,
double (* f) (const gsl_vector * x, void * params)this function should return the result
f(x,params) for argument x and parameters params.
If the function cannot be computed, an error value of GSL_NAN
should be returned.
size_t nthe dimension of the system, i.e. the number of components of the vectors x.
void * paramsa pointer to the parameters of the function.
The following example function defines a simple two-dimensional paraboloid with five parameters,
/* Paraboloid centered on (p[0],p[1]), with
scale factors (p[2],p[3]) and minimum p[4] */
double
my_f (const gsl_vector *v, void *params)
{
double x, y;
double *p = (double *)params;
x = gsl_vector_get(v, 0);
y = gsl_vector_get(v, 1);
return p[2] * (x - p[0]) * (x - p[0]) +
p[3] * (y - p[1]) * (y - p[1]) + p[4];
}
/* The gradient of f, df = (df/dx, df/dy). */
void
my_df (const gsl_vector *v, void *params,
gsl_vector *df)
{
double x, y;
double *p = (double *)params;
x = gsl_vector_get(v, 0);
y = gsl_vector_get(v, 1);
gsl_vector_set(df, 0, 2.0 * p[2] * (x - p[0]));
gsl_vector_set(df, 1, 2.0 * p[3] * (y - p[1]));
}
/* Compute both f and df together. */
void
my_fdf (const gsl_vector *x, void *params,
double *f, gsl_vector *df)
{
*f = my_f(x, params);
my_df(x, params, df);
}
The function can be initialized using the following code,
gsl_multimin_function_fdf my_func;
/* Paraboloid center at (1,2), scale factors (10, 20),
minimum value 30 */
double p[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 };
my_func.n = 2; /* number of function components */
my_func.f = &my_f;
my_func.df = &my_df;
my_func.fdf = &my_fdf;
my_func.params = (void *)p;
Next: Multimin Iteration, Previous: Initializing the Multidimensional Minimizer, Up: Multidimensional Minimization [Index]
Next: Zeros of Airy Functions, Previous: Airy Functions, Up: Airy Functions and Derivatives [Index]
These routines compute the Airy function derivative Ai'(x) with an accuracy specified by mode.
These routines compute the Airy function derivative Bi'(x) with an accuracy specified by mode.
These routines compute the scaled Airy function derivative S_A(x) Ai'(x). For x>0 the scaling factor S_A(x) is \exp(+(2/3) x^(3/2)), and is 1 for x<0.
These routines compute the scaled Airy function derivative S_B(x) Bi'(x). For x>0 the scaling factor S_B(x) is exp(-(2/3) x^(3/2)), and is 1 for x<0.
Next: The Exponential Power Distribution, Previous: The Exponential Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the Laplace distribution with width a. The distribution is,
p(x) dx = {1 \over 2 a} \exp(-|x/a|) dx
for -\infty < x < \infty.
This function computes the probability density p(x) at x for a Laplace distribution with width a, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Laplace distribution with width a.
Next: The Gaussian Tail Distribution, Previous: Random Number Distribution Introduction, Up: Random Number Distributions [Index]
This function returns a Gaussian random variate, with mean zero and standard deviation sigma. The probability distribution for Gaussian random variates is,
p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx
for x in the range -\infty to +\infty. Use the
transformation z = \mu + x on the numbers returned by
gsl_ran_gaussian to obtain a Gaussian distribution with mean
\mu. This function uses the Box-Muller algorithm which requires two
calls to the random number generator r.
This function computes the probability density p(x) at x for a Gaussian distribution with standard deviation sigma, using the formula given above.
This function computes a Gaussian random variate using the alternative Marsaglia-Tsang ziggurat and Kinderman-Monahan-Leva ratio methods. The Ziggurat algorithm is the fastest available algorithm in most cases.
These functions compute results for the unit Gaussian distribution. They are equivalent to the functions above with a standard deviation of one, sigma = 1.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Gaussian distribution with standard deviation sigma.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the unit Gaussian distribution.
Next: The Gaussian Tail Distribution, Previous: Random Number Distribution Introduction, Up: Random Number Distributions [Index]
Next: Complex Hyperbolic Functions, Previous: Complex Trigonometric Functions, Up: Complex Numbers [Index]
This function returns the complex arcsine of the complex number z, \arcsin(z). The branch cuts are on the real axis, less than -1 and greater than 1.
This function returns the complex arcsine of the real number z, \arcsin(z). For z between -1 and 1, the function returns a real value in the range [-\pi/2,\pi/2]. For z less than -1 the result has a real part of -\pi/2 and a positive imaginary part. For z greater than 1 the result has a real part of \pi/2 and a negative imaginary part.
This function returns the complex arccosine of the complex number z, \arccos(z). The branch cuts are on the real axis, less than -1 and greater than 1.
This function returns the complex arccosine of the real number z, \arccos(z). For z between -1 and 1, the function returns a real value in the range [0,\pi]. For z less than -1 the result has a real part of \pi and a negative imaginary part. For z greater than 1 the result is purely imaginary and positive.
This function returns the complex arctangent of the complex number z, \arctan(z). The branch cuts are on the imaginary axis, below -i and above i.
This function returns the complex arcsecant of the complex number z, \arcsec(z) = \arccos(1/z).
This function returns the complex arcsecant of the real number z, \arcsec(z) = \arccos(1/z).
This function returns the complex arccosecant of the complex number z, \arccsc(z) = \arcsin(1/z).
This function returns the complex arccosecant of the real number z, \arccsc(z) = \arcsin(1/z).
This function returns the complex arccotangent of the complex number z, \arccot(z) = \arctan(1/z).
Next: Complex Hyperbolic Functions, Previous: Complex Trigonometric Functions, Up: Complex Numbers [Index]
Next: Random Number Acknowledgements, Previous: Random Number Generator Examples, Up: Random Number Generation [Index]
The subject of random number generation and testing is reviewed extensively in Knuth’s Seminumerical Algorithms.
Further information is available in the review paper written by Pierre L’Ecuyer,
http://www.iro.umontreal.ca/~lecuyer/papers.html in the file handsim.ps.
The source code for the DIEHARD random number generator tests is also available online,
A comprehensive set of random number generator tests is available from NIST,
Next: Incomplete Fermi-Dirac Integrals, Up: Fermi-Dirac Function [Index]
The complete Fermi-Dirac integral F_j(x) is given by,
F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))
Note that the Fermi-Dirac integral is sometimes defined without the normalisation factor in other texts.
These routines compute the complete Fermi-Dirac integral with an index of -1. This integral is given by F_{-1}(x) = e^x / (1 + e^x).
These routines compute the complete Fermi-Dirac integral with an index of 0. This integral is given by F_0(x) = \ln(1 + e^x).
These routines compute the complete Fermi-Dirac integral with an index of 1, F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1)).
These routines compute the complete Fermi-Dirac integral with an index of 2, F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1)).
These routines compute the complete Fermi-Dirac integral with an integer index of j, F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j /(\exp(t-x)+1)).
These routines compute the complete Fermi-Dirac integral F_{-1/2}(x).
These routines compute the complete Fermi-Dirac integral F_{1/2}(x).
These routines compute the complete Fermi-Dirac integral F_{3/2}(x).
Next: Incomplete Fermi-Dirac Integrals, Up: Fermi-Dirac Function [Index]
A Chebyshev series is stored using the following structure,
typedef struct
{
double * c; /* coefficients c[0] .. c[order] */
int order; /* order of expansion */
double a; /* lower interval point */
double b; /* upper interval point */
...
} gsl_cheb_series
The approximation is made over the range [a,b] using order+1 terms, including the coefficient c[0]. The series is computed using the following convention,
f(x) = (c_0 / 2) + \sum_{n=1} c_n T_n(x)
which is needed when accessing the coefficients directly.
gsl-ref-html-2.3/Opening-an-existing-ntuple-file.html 0000664 0001750 0001750 00000010103 13055414474 020670 0 ustar edd eddNext: Writing ntuples, Previous: Creating ntuples, Up: N-tuples [Index]
This function opens an existing ntuple file filename for reading and returns a pointer to a corresponding ntuple struct. The ntuples in the file must have size size. A pointer to memory for the current ntuple row ntuple_data must be supplied—this is used to copy ntuples in and out of the file.
Previous: Further Information, Up: Introduction [Index]
This manual contains many examples which can be typed at the keyboard. A command entered at the terminal is shown like this,
$ command
The first character on the line is the terminal prompt, and should not be typed. The dollar sign ‘$’ is used as the standard prompt in this manual, although some systems may use a different character.
The examples assume the use of the GNU operating system. There may be
minor differences in the output on other systems. The commands for
setting environment variables use the Bourne shell syntax of the
standard GNU shell (bash).
Next: IEEE References and Further Reading, Previous: Representation of floating point numbers, Up: IEEE floating-point arithmetic [Index]
The IEEE standard defines several modes for controlling the behavior of floating point operations. These modes specify the important properties of computer arithmetic: the direction used for rounding (e.g. whether numbers should be rounded up, down or to the nearest number), the rounding precision and how the program should handle arithmetic exceptions, such as division by zero.
Many of these features can now be controlled via standard functions such
as fpsetround, which should be used whenever they are available.
Unfortunately in the past there has been no universal API for
controlling their behavior—each system has had its own low-level way
of accessing them. To help you write portable programs GSL allows you
to specify modes in a platform-independent way using the environment
variable GSL_IEEE_MODE. The library then takes care of all the
necessary machine-specific initializations for you when you call the
function gsl_ieee_env_setup.
This function reads the environment variable GSL_IEEE_MODE and
attempts to set up the corresponding specified IEEE modes. The
environment variable should be a list of keywords, separated by
commas, like this,
GSL_IEEE_MODE = "keyword,keyword,..."
where keyword is one of the following mode-names,
single-precision
double-precision
extended-precision
round-to-nearest
round-down
round-up
round-to-zero
mask-all
mask-invalid
mask-denormalized
mask-division-by-zero
mask-overflow
mask-underflow
trap-inexact
trap-common
If GSL_IEEE_MODE is empty or undefined then the function returns
immediately and no attempt is made to change the system’s IEEE
mode. When the modes from GSL_IEEE_MODE are turned on the
function prints a short message showing the new settings to remind you
that the results of the program will be affected.
If the requested modes are not supported by the platform being used then
the function calls the error handler and returns an error code of
GSL_EUNSUP.
When options are specified using this method, the resulting mode is
based on a default setting of the highest available precision (double
precision or extended precision, depending on the platform) in
round-to-nearest mode, with all exceptions enabled apart from the
INEXACT exception. The INEXACT exception is generated
whenever rounding occurs, so it must generally be disabled in typical
scientific calculations. All other floating-point exceptions are
enabled by default, including underflows and the use of denormalized
numbers, for safety. They can be disabled with the individual
mask- settings or together using mask-all.
The following adjusted combination of modes is convenient for many purposes,
GSL_IEEE_MODE="double-precision,"\
"mask-underflow,"\
"mask-denormalized"
This choice ignores any errors relating to small numbers (either denormalized, or underflowing to zero) but traps overflows, division by zero and invalid operations.
Note that on the x86 series of processors this function sets both the original x87 mode and the newer MXCSR mode, which controls SSE floating-point operations. The SSE floating-point units do not have a precision-control bit, and always work in double-precision. The single-precision and extended-precision keywords have no effect in this case.
To demonstrate the effects of different rounding modes consider the following program which computes e, the base of natural logarithms, by summing a rapidly-decreasing series,
e = 1 + 1/2! + 1/3! + 1/4! + ... = 2.71828182846...
#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_ieee_utils.h>
int
main (void)
{
double x = 1, oldsum = 0, sum = 0;
int i = 0;
gsl_ieee_env_setup (); /* read GSL_IEEE_MODE */
do
{
i++;
oldsum = sum;
sum += x;
x = x / i;
printf ("i=%2d sum=%.18f error=%g\n",
i, sum, sum - M_E);
if (i > 30)
break;
}
while (sum != oldsum);
return 0;
}
Here are the results of running the program in round-to-nearest
mode. This is the IEEE default so it isn’t really necessary to specify
it here,
$ GSL_IEEE_MODE="round-to-nearest" ./a.out i= 1 sum=1.000000000000000000 error=-1.71828 i= 2 sum=2.000000000000000000 error=-0.718282 .... i=18 sum=2.718281828459045535 error=4.44089e-16 i=19 sum=2.718281828459045535 error=4.44089e-16
After nineteen terms the sum converges to within 4 \times 10^-16 of the correct value.
If we now change the rounding mode to
round-down the final result is less accurate,
$ GSL_IEEE_MODE="round-down" ./a.out i= 1 sum=1.000000000000000000 error=-1.71828 .... i=19 sum=2.718281828459041094 error=-3.9968e-15
The result is about
4 \times 10^-15
below the correct value, an order of magnitude worse than the result
obtained in the round-to-nearest mode.
If we change to rounding mode to round-up then the final result
is higher than the correct value (when we add each term to the sum the
final result is always rounded up, which increases the sum by at least
one tick until the added term underflows to zero). To avoid this
problem we would need to use a safer converge criterion, such as
while (fabs(sum - oldsum) > epsilon), with a suitably chosen
value of epsilon.
Finally we can see the effect of computing the sum using
single-precision rounding, in the default round-to-nearest
mode. In this case the program thinks it is still using double precision
numbers but the CPU rounds the result of each floating point operation
to single-precision accuracy. This simulates the effect of writing the
program using single-precision float variables instead of
double variables. The iteration stops after about half the number
of iterations and the final result is much less accurate,
$ GSL_IEEE_MODE="single-precision" ./a.out .... i=12 sum=2.718281984329223633 error=1.5587e-07
with an error of O(10^-7), which corresponds to single precision accuracy (about 1 part in 10^7). Continuing the iterations further does not decrease the error because all the subsequent results are rounded to the same value.
Next: IEEE References and Further Reading, Previous: Representation of floating point numbers, Up: IEEE floating-point arithmetic [Index]
Next: Maximum and Minimum values, Previous: Correlation, Up: Statistics [Index]
The functions described in this section allow the computation of statistics for weighted samples. The functions accept an array of samples, x_i, with associated weights, w_i. Each sample x_i is considered as having been drawn from a Gaussian distribution with variance \sigma_i^2. The sample weight w_i is defined as the reciprocal of this variance, w_i = 1/\sigma_i^2. Setting a weight to zero corresponds to removing a sample from a dataset.
This function returns the weighted mean of the dataset data with stride stride and length n, using the set of weights w with stride wstride and length n. The weighted mean is defined as,
\Hat\mu = (\sum w_i x_i) / (\sum w_i)
This function returns the estimated variance of the dataset data with stride stride and length n, using the set of weights w with stride wstride and length n. The estimated variance of a weighted dataset is calculated as,
\Hat\sigma^2 = ((\sum w_i)/((\sum w_i)^2 - \sum (w_i^2)))
\sum w_i (x_i - \Hat\mu)^2
Note that this expression reduces to an unweighted variance with the familiar 1/(N-1) factor when there are N equal non-zero weights.
This function returns the estimated variance of the weighted dataset data using the given weighted mean wmean.
The standard deviation is defined as the square root of the variance.
This function returns the square root of the corresponding variance
function gsl_stats_wvariance above.
This function returns the square root of the corresponding variance
function gsl_stats_wvariance_m above.
This function computes an unbiased estimate of the variance of the weighted dataset data when the population mean mean of the underlying distribution is known a priori. In this case the estimator for the variance replaces the sample mean \Hat\mu by the known population mean \mu,
\Hat\sigma^2 = (\sum w_i (x_i - \mu)^2) / (\sum w_i)
The standard deviation is defined as the square root of the variance. This function returns the square root of the corresponding variance function above.
These functions return the weighted total sum of squares (TSS) of
data about the weighted mean. For gsl_stats_wtss_m the
user-supplied value of wmean is used, and for gsl_stats_wtss
it is computed using gsl_stats_wmean.
TSS = \sum w_i (x_i - wmean)^2
This function computes the weighted absolute deviation from the weighted mean of data. The absolute deviation from the mean is defined as,
absdev = (\sum w_i |x_i - \Hat\mu|) / (\sum w_i)
This function computes the absolute deviation of the weighted dataset data about the given weighted mean wmean.
This function computes the weighted skewness of the dataset data.
skew = (\sum w_i ((x_i - \Hat x)/\Hat \sigma)^3) / (\sum w_i)
This function computes the weighted skewness of the dataset data using the given values of the weighted mean and weighted standard deviation, wmean and wsd.
This function computes the weighted kurtosis of the dataset data.
kurtosis = ((\sum w_i ((x_i - \Hat x)/\Hat \sigma)^4) / (\sum w_i)) - 3
This function computes the weighted kurtosis of the dataset data using the given values of the weighted mean and weighted standard deviation, wmean and wsd.
Next: Maximum and Minimum values, Previous: Correlation, Up: Statistics [Index]
Next: PLAIN Monte Carlo, Up: Monte Carlo Integration [Index]
All of the Monte Carlo integration routines use the same general form of interface. There is an allocator to allocate memory for control variables and workspace, a routine to initialize those control variables, the integrator itself, and a function to free the space when done.
Each integration function requires a random number generator to be supplied, and returns an estimate of the integral and its standard deviation. The accuracy of the result is determined by the number of function calls specified by the user. If a known level of accuracy is required this can be achieved by calling the integrator several times and averaging the individual results until the desired accuracy is obtained.
Random sample points used within the Monte Carlo routines are always chosen strictly within the integration region, so that endpoint singularities are automatically avoided.
The function to be integrated has its own datatype, defined in the header file gsl_monte.h.
This data type defines a general function with parameters for Monte Carlo integration.
double (* f) (double * x, size_t dim, void * params)this function should return the value f(x,params) for the argument x and parameters params, where x is an array of size dim giving the coordinates of the point where the function is to be evaluated.
size_t dimthe number of dimensions for x.
void * paramsa pointer to the parameters of the function.
Here is an example for a quadratic function in two dimensions,
f(x,y) = a x^2 + b x y + c y^2
with a = 3, b = 2, c = 1. The following code
defines a gsl_monte_function F which you could pass to an
integrator:
struct my_f_params { double a; double b; double c; };
double
my_f (double x[], size_t dim, void * p) {
struct my_f_params * fp = (struct my_f_params *)p;
if (dim != 2)
{
fprintf (stderr, "error: dim != 2");
abort ();
}
return fp->a * x[0] * x[0]
+ fp->b * x[0] * x[1]
+ fp->c * x[1] * x[1];
}
gsl_monte_function F;
struct my_f_params params = { 3.0, 2.0, 1.0 };
F.f = &my_f;
F.dim = 2;
F.params = ¶ms;
The function f(x) can be evaluated using the following macro,
#define GSL_MONTE_FN_EVAL(F,x)
(*((F)->f))(x,(F)->dim,(F)->params)
Next: PLAIN Monte Carlo, Up: Monte Carlo Integration [Index]
Next: Error Reporting Examples, Previous: Error Handlers, Up: Error Handling [Index]
If you are writing numerical functions in a program which also uses GSL code you may find it convenient to adopt the same error reporting conventions as in the library.
To report an error you need to call the function gsl_error with a
string describing the error and then return an appropriate error code
from gsl_errno.h, or a special value, such as NaN. For
convenience the file gsl_errno.h defines two macros which carry
out these steps:
This macro reports an error using the GSL conventions and returns a
status value of gsl_errno. It expands to the following code fragment,
gsl_error (reason, __FILE__, __LINE__, gsl_errno); return gsl_errno;
The macro definition in gsl_errno.h actually wraps the code
in a do { ... } while (0) block to prevent possible
parsing problems.
Here is an example of how the macro could be used to report that a
routine did not achieve a requested tolerance. To report the error the
routine needs to return the error code GSL_ETOL.
if (residual > tolerance)
{
GSL_ERROR("residual exceeds tolerance", GSL_ETOL);
}
This macro is the same as GSL_ERROR but returns a user-defined
value of value instead of an error code. It can be used for
mathematical functions that return a floating point value.
The following example shows how to return a NaN at a mathematical
singularity using the GSL_ERROR_VAL macro,
if (x == 0)
{
GSL_ERROR_VAL("argument lies on singularity",
GSL_ERANGE, GSL_NAN);
}
Next: Error Reporting Examples, Previous: Error Handlers, Up: Error Handling [Index]
Next: Nonlinear Least-Squares Weighted Overview, Previous: Nonlinear Least-Squares Overview, Up: Nonlinear Least-Squares Fitting [Index]
Below we describe the methods available for solving the trust region
subproblem. The methods available provide either exact or approximate
solutions to the trust region subproblem. In all algorithms below,
the Hessian matrix B_k is approximated as B_k \approx J_k^T J_k,
where J_k = J(x_k). In all methods, the solution of the TRS
involves solving a linear least squares system involving the Jacobian
matrix. For small to moderate sized problems (gsl_multifit_nlinear interface),
this is accomplished by factoring the full Jacobian matrix, which is provided
by the user, with the Cholesky, QR, or SVD decompositions. For large systems
(gsl_multilarge_nlinear interface), the user has two choices. One
is to solve the system iteratively, without needing to store the full
Jacobian matrix in memory. With this method, the user must provide a routine
to calculate the matrix-vector products J u or J^T u for a given vector u.
This iterative method is particularly useful for systems where the Jacobian has
sparse structure, since forming matrix-vector products can be done cheaply. The
second option for large systems involves forming the normal equations matrix
J^T J and then factoring it using a Cholesky decomposition. The normal
equations matrix is p-by-p, typically much smaller than the full
n-by-p Jacobian, and can usually be stored in memory even if the full
Jacobian matrix cannot. This option is useful for large, dense systems, or if the
iterative method has difficulty converging.
Next: Nonlinear Least-Squares Weighted Overview, Previous: Nonlinear Least-Squares Overview, Up: Nonlinear Least-Squares Fitting [Index]
Previous: Maximum and Minimum functions, Up: Mathematical Functions [Index]
It is sometimes useful to be able to compare two floating point numbers approximately, to allow for rounding and truncation errors. The following function implements the approximate floating-point comparison algorithm proposed by D.E. Knuth in Section 4.2.2 of Seminumerical Algorithms (3rd edition).
This function determines whether x and y are approximately equal to a relative accuracy epsilon.
The relative accuracy is measured using an interval of size 2
\delta, where \delta = 2^k \epsilon and k is the
maximum base-2 exponent of x and y as computed by the
function frexp.
If x and y lie within this interval, they are considered approximately equal and the function returns 0. Otherwise if x < y, the function returns -1, or if x > y, the function returns +1.
Note that x and y are compared to relative accuracy, so this function is not suitable for testing whether a value is approximately zero.
The implementation is based on the package fcmp by T.C. Belding.
Previous: Zeros of Airy Functions, Up: Airy Functions and Derivatives [Index]
These routines compute the location of the s-th zero of the Airy function derivative Ai'(x).
These routines compute the location of the s-th zero of the Airy function derivative Bi'(x).
Next: Reporting Bugs, Previous: Obtaining GSL, Up: Introduction [Index]
The software described in this manual has no warranty, it is provided “as is”. It is your responsibility to validate the behavior of the routines and their accuracy using the source code provided, or to purchase support and warranties from commercial redistributors. Consult the GNU General Public license for further details (see GNU General Public License).
gsl-ref-html-2.3/Eigensystems.html 0000664 0001750 0001750 00000015023 13055414420 015277 0 ustar edd eddNext: Fast Fourier Transforms, Previous: Linear Algebra, Up: Top [Index]
This chapter describes functions for computing eigenvalues and eigenvectors of matrices. There are routines for real symmetric, real nonsymmetric, complex hermitian, real generalized symmetric-definite, complex generalized hermitian-definite, and real generalized nonsymmetric eigensystems. Eigenvalues can be computed with or without eigenvectors. The hermitian and real symmetric matrix algorithms are symmetric bidiagonalization followed by QR reduction. The nonsymmetric algorithm is the Francis QR double-shift. The generalized nonsymmetric algorithm is the QZ method due to Moler and Stewart.
The functions described in this chapter are declared in the header file gsl_eigen.h.
Next: Copying 2D Histograms, Previous: The 2D histogram struct, Up: Histograms [Index]
The functions for allocating memory to a 2D histogram follow the style
of malloc and free. In addition they also perform their
own error checking. If there is insufficient memory available to
allocate a histogram then the functions call the error handler (with
an error number of GSL_ENOMEM) in addition to returning a null
pointer. Thus if you use the library error handler to abort your program
then it isn’t necessary to check every 2D histogram alloc.
This function allocates memory for a two-dimensional histogram with
nx bins in the x direction and ny bins in the y direction.
The function returns a pointer to a newly created gsl_histogram2d
struct. If insufficient memory is available a null pointer is returned
and the error handler is invoked with an error code of
GSL_ENOMEM. The bins and ranges must be initialized with one of
the functions below before the histogram is ready for use.
This function sets the ranges of the existing histogram h using the arrays xrange and yrange of size xsize and ysize respectively. The values of the histogram bins are reset to zero.
This function sets the ranges of the existing histogram h to cover the ranges xmin to xmax and ymin to ymax uniformly. The values of the histogram bins are reset to zero.
This function frees the 2D histogram h and all of the memory associated with it.
Next: Copying 2D Histograms, Previous: The 2D histogram struct, Up: Histograms [Index]
Next: Other random number generators, Previous: Random number generator algorithms, Up: Random Number Generation [Index]
The standard Unix random number generators rand, random
and rand48 are provided as part of GSL. Although these
generators are widely available individually often they aren’t all
available on the same platform. This makes it difficult to write
portable code using them and so we have included the complete set of
Unix generators in GSL for convenience. Note that these generators
don’t produce high-quality randomness and aren’t suitable for work
requiring accurate statistics. However, if you won’t be measuring
statistical quantities and just want to introduce some variation into
your program then these generators are quite acceptable.
This is the BSD rand generator. Its sequence is
x_{n+1} = (a x_n + c) mod m
with a = 1103515245, c = 12345 and m = 2^31. The seed specifies the initial value, x_1. The period of this generator is 2^31, and it uses 1 word of storage per generator.
These generators implement the random family of functions, a
set of linear feedback shift register generators originally used in BSD
Unix. There are several versions of random in use today: the
original BSD version (e.g. on SunOS4), a libc5 version (found on
older GNU/Linux systems) and a glibc2 version. Each version uses a
different seeding procedure, and thus produces different sequences.
The original BSD routines accepted a variable length buffer for the
generator state, with longer buffers providing higher-quality
randomness. The random function implemented algorithms for
buffer lengths of 8, 32, 64, 128 and 256 bytes, and the algorithm with
the largest length that would fit into the user-supplied buffer was
used. To support these algorithms additional generators are available
with the following names,
gsl_rng_random8_bsd gsl_rng_random32_bsd gsl_rng_random64_bsd gsl_rng_random128_bsd gsl_rng_random256_bsd
where the numeric suffix indicates the buffer length. The original BSD
random function used a 128-byte default buffer and so
gsl_rng_random_bsd has been made equivalent to
gsl_rng_random128_bsd. Corresponding versions of the libc5
and glibc2 generators are also available, with the names
gsl_rng_random8_libc5, gsl_rng_random8_glibc2, etc.
This is the Unix rand48 generator. Its sequence is
x_{n+1} = (a x_n + c) mod m
defined on 48-bit unsigned integers with
a = 25214903917,
c = 11 and
m = 2^48.
The seed specifies the upper 32 bits of the initial value, x_1,
with the lower 16 bits set to 0x330E. The function
gsl_rng_get returns the upper 32 bits from each term of the
sequence. This does not have a direct parallel in the original
rand48 functions, but forcing the result to type long int
reproduces the output of mrand48. The function
gsl_rng_uniform uses the full 48 bits of internal state to return
the double precision number x_n/m, which is equivalent to the
function drand48. Note that some versions of the GNU C Library
contained a bug in mrand48 function which caused it to produce
different results (only the lower 16-bits of the return value were set).
Next: Other random number generators, Previous: Random number generator algorithms, Up: Random Number Generation [Index]
Next: The Dirichlet Distribution, Previous: The Type-1 Gumbel Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the Type-2 Gumbel distribution. The Type-2 Gumbel distribution function is,
p(x) dx = a b x^{-a-1} \exp(-b x^{-a}) dx
for 0 < x < \infty.
This function computes the probability density p(x) at x for a Type-2 Gumbel distribution with parameters a and b, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Type-2 Gumbel distribution with parameters a and b.
Next: Code Reuse, Previous: Thread-safety, Up: Using the library [Index]
From time to time, it may be necessary for the definitions of some
functions to be altered or removed from the library. In these
circumstances the functions will first be declared deprecated and
then removed from subsequent versions of the library. Functions that
are deprecated can be disabled in the current release by setting the
preprocessor definition GSL_DISABLE_DEPRECATED. This allows
existing code to be tested for forwards compatibility.
Next: Nonlinear Least-Squares Iteration, Previous: Nonlinear Least-Squares Initialization, Up: Nonlinear Least-Squares Fitting [Index]
The user must provide n functions of p variables for the minimization algorithm to operate on. In order to allow for arbitrary parameters the functions are defined by the following data types:
This data type defines a general system of functions with arbitrary parameters, the corresponding Jacobian matrix of derivatives, and optionally the second directional derivative of the functions for geodesic acceleration.
int (* f) (const gsl_vector * x, void * params, gsl_vector * f)This function should store the n components of the vector f(x) in f for argument x and arbitrary parameters params, returning an appropriate error code if the function cannot be computed.
int (* df) (const gsl_vector * x, void * params, gsl_matrix * J)This function should store the n-by-p matrix result J_ij = d f_i(x) / d x_j in J for argument x and arbitrary parameters params, returning an appropriate error code if the matrix cannot be computed. If an analytic Jacobian is unavailable, or too expensive to compute, this function pointer may be set to NULL, in which case the Jacobian will be internally computed using finite difference approximations of the function f.
int (* fvv) (const gsl_vector * x, const gsl_vector * v, void * params, gsl_vector * fvv)When geodesic acceleration is enabled, this function should store the n components of the vector f_{vv}(x) = \sum_{\alpha\beta} v_{\alpha} v_{\beta} {\partial \over \partial x_{\alpha}} {\partial \over \partial x_{\beta}} f(x), representing second directional derivatives of the function to be minimized, into the output fvv. The parameter vector is provided in x and the velocity vector is provided in v, both of which have p components. The arbitrary parameters are given in params. If analytic expressions for f_{vv}(x) are unavailable or too difficult to compute, this function pointer may be set to NULL, in which case f_{vv}(x) will be computed internally using a finite difference approximation.
size_t nthe number of functions, i.e. the number of components of the vector f.
size_t pthe number of independent variables, i.e. the number of components of the vector x.
void * paramsa pointer to the arbitrary parameters of the function.
size_t nevalfThis does not need to be set by the user. It counts the number of
function evaluations and is initialized by the _init function.
size_t nevaldfThis does not need to be set by the user. It counts the number of
Jacobian evaluations and is initialized by the _init function.
size_t nevalfvvThis does not need to be set by the user. It counts the number of
f_{vv}(x) evaluations and is initialized by the _init function.
This data type defines a general system of functions with arbitrary parameters, a function to compute J u or J^T u for a given vector u, the normal equations matrix J^T J, and optionally the second directional derivative of the functions for geodesic acceleration.
int (* f) (const gsl_vector * x, void * params, gsl_vector * f)This function should store the n components of the vector f(x) in f for argument x and arbitrary parameters params, returning an appropriate error code if the function cannot be computed.
int (* df) (CBLAS_TRANSPOSE_t TransJ, const gsl_vector * x, const gsl_vector * u, void * params, gsl_vector * v, gsl_matrix * JTJ)If TransJ is equal to CblasNoTrans, then this function should
compute the matrix-vector product J u and store the result in v.
If TransJ is equal to CblasTrans, then this function should
compute the matrix-vector product J^T u and store the result in v.
Additionally, the normal equations matrix J^T J should be stored in the
lower half of JTJ. The input matrix JTJ could be set to NULL,
for example by iterative methods which do not require this matrix, so the user
should check for this prior to constructing the matrix.
The input params contains the arbitrary parameters.
int (* fvv) (const gsl_vector * x, const gsl_vector * v, void * params, gsl_vector * fvv)When geodesic acceleration is enabled, this function should store the n components of the vector f_{vv}(x) = \sum_{\alpha\beta} v_{\alpha} v_{\beta} {\partial \over \partial x_{\alpha}} {\partial \over \partial x_{\beta}} f(x), representing second directional derivatives of the function to be minimized, into the output fvv. The parameter vector is provided in x and the velocity vector is provided in v, both of which have p components. The arbitrary parameters are given in params. If analytic expressions for f_{vv}(x) are unavailable or too difficult to compute, this function pointer may be set to NULL, in which case f_{vv}(x) will be computed internally using a finite difference approximation.
size_t nthe number of functions, i.e. the number of components of the vector f.
size_t pthe number of independent variables, i.e. the number of components of the vector x.
void * paramsa pointer to the arbitrary parameters of the function.
size_t nevalfThis does not need to be set by the user. It counts the number of
function evaluations and is initialized by the _init function.
size_t nevaldfuThis does not need to be set by the user. It counts the number of
Jacobian matrix-vector evaluations (J u or J^T u) and
is initialized by the _init function.
size_t nevaldf2This does not need to be set by the user. It counts the number of
J^T J evaluations and is initialized by the _init function.
size_t nevalfvvThis does not need to be set by the user. It counts the number of
f_{vv}(x) evaluations and is initialized by the _init function.
Note that when fitting a non-linear model against experimental data, the data is passed to the functions above using the params argument and the trial best-fit parameters through the x argument.
Next: Nonlinear Least-Squares Iteration, Previous: Nonlinear Least-Squares Initialization, Up: Nonlinear Least-Squares Fitting [Index]
Next: Sparse BLAS Support, Previous: Basis Splines, Up: Top [Index]
This chapter describes functions for the construction and manipulation of sparse matrices, matrices which are populated primarily with zeros and contain only a few non-zero elements. Sparse matrices often appear in the solution of partial differential equations. It is beneficial to use specialized data structures and algorithms for storing and working with sparse matrices, since dense matrix algorithms and structures can be very slow and use huge amounts of memory when applied to sparse matrices.
The header file gsl_spmatrix.h contains the prototypes for the sparse matrix functions and related declarations.
Next: Sparse BLAS Support, Previous: Basis Splines, Up: Top [Index]
Next: BLAS Examples, Up: BLAS Support [Index]
GSL provides dense vector and matrix objects, based on the relevant built-in types. The library provides an interface to the BLAS operations which apply to these objects. The interface to this functionality is given in the file gsl_blas.h.
| • Level 1 GSL BLAS Interface: | ||
| • Level 2 GSL BLAS Interface: | ||
| • Level 3 GSL BLAS Interface: |
Next: The Rayleigh Tail Distribution, Previous: The Cauchy Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the Rayleigh distribution with scale parameter sigma. The distribution is,
p(x) dx = {x \over \sigma^2} \exp(- x^2/(2 \sigma^2)) dx
for x > 0.
This function computes the probability density p(x) at x for a Rayleigh distribution with scale parameter sigma, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Rayleigh distribution with scale parameter sigma.
Previous: Nonlinear Least-Squares Examples, Up: Nonlinear Least-Squares Fitting [Index]
The following publications are relevant to the algorithms described in this section,
Next: Derivatives and Integrals, Previous: Auxiliary Functions for Chebyshev Series, Up: Chebyshev Approximations [Index]
This function evaluates the Chebyshev series cs at a given point x.
This function computes the Chebyshev series cs at a given point x, estimating both the series result and its absolute error abserr. The error estimate is made from the first neglected term in the series.
This function evaluates the Chebyshev series cs at a given point x, to (at most) the given order order.
This function evaluates a Chebyshev series cs at a given point x, estimating both the series result and its absolute error abserr, to (at most) the given order order. The error estimate is made from the first neglected term in the series.
Next: The Flat (Uniform) Distribution, Previous: The Levy skew alpha-Stable Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the gamma distribution. The distribution function is,
p(x) dx = {1 \over \Gamma(a) b^a} x^{a-1} e^{-x/b} dx
for x > 0.
The gamma distribution with an integer parameter a is known as the Erlang distribution.
The variates are computed using the Marsaglia-Tsang fast gamma method.
This function for this method was previously called
gsl_ran_gamma_mt and can still be accessed using this name.
This function returns a gamma variate using the algorithms from Knuth (vol 2).
This function computes the probability density p(x) at x for a gamma distribution with parameters a and b, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the gamma distribution with parameters a and b.
Next: Nonlinear Least-Squares TRS Dogleg, Previous: Nonlinear Least-Squares TRS Levenberg-Marquardt, Up: Nonlinear Least-Squares TRS Overview [Index]
This method applies a so-called geodesic acceleration correction to the standard Levenberg-Marquardt step \delta_k (Transtrum et al, 2011). By interpreting \delta_k as a first order step along a geodesic in the model parameter space (ie: a velocity \delta_k = v_k), the geodesic acceleration a_k is a second order correction along the geodesic which is determined by solving the linear least squares system
[J_k; sqrt(mu_k) D_k] a_k = - [f_vv(x_k); 0]
where f_{vv} is the second directional derivative of
the residual vector in the velocity direction v,
f_{vv}(x) = D_v^2 f = \sum_{\alpha\beta} v_{\alpha} v_{\beta} \partial_{\alpha} \partial_{\beta} f(x),
where \alpha and \beta are summed over the p
parameters. The new total step is then \delta_k' = v_k + {1 \over 2}a_k.
The second order correction a_k can be calculated with a modest additional
cost, and has been shown to dramatically reduce the number of iterations
(and expensive Jacobian evaluations) required to reach convergence on a variety
of different problems. In order to utilize the geodesic acceleration, the user must supply a
function which provides the second directional derivative vector
f_{vv}(x), or alternatively the library can use a finite
difference method to estimate this vector with one additional function
evaluation of f(x + h v) where h is a tunable step size
(see the h_fvv parameter description).
Next: QR Decomposition with Column Pivoting, Previous: LU Decomposition, Up: Linear Algebra [Index]
A general rectangular M-by-N matrix A has a QR decomposition into the product of an orthogonal M-by-M square matrix Q (where Q^T Q = I) and an M-by-N right-triangular matrix R,
A = Q R
This decomposition can be used to convert the linear system A x = b into the triangular system R x = Q^T b, which can be solved by back-substitution. Another use of the QR decomposition is to compute an orthonormal basis for a set of vectors. The first N columns of Q form an orthonormal basis for the range of A, ran(A), when A has full column rank.
This function factorizes the M-by-N matrix A into the QR decomposition A = Q R. On output the diagonal and upper triangular part of the input matrix contain the matrix R. The vector tau and the columns of the lower triangular part of the matrix A contain the Householder coefficients and Householder vectors which encode the orthogonal matrix Q. The vector tau must be of length k=\min(M,N). The matrix Q is related to these components by, Q = Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i v_i^T and v_i is the Householder vector v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)). This is the same storage scheme as used by LAPACK.
The algorithm used to perform the decomposition is Householder QR (Golub & Van Loan, Matrix Computations, Algorithm 5.2.1).
This function solves the square system A x = b using the QR
decomposition of A held in (QR, tau) which must
have been computed previously with gsl_linalg_QR_decomp.
The least-squares solution for
rectangular systems can be found using gsl_linalg_QR_lssolve.
This function solves the square system A x = b in-place using
the QR decomposition of A held in (QR,tau)
which must have been computed previously by
gsl_linalg_QR_decomp. On input x should contain the
right-hand side b, which is replaced by the solution on output.
This function finds the least squares solution to the overdetermined
system A x = b where the matrix A has more rows than
columns. The least squares solution minimizes the Euclidean norm of the
residual, ||Ax - b||.The routine requires as input
the QR decomposition
of A into (QR, tau) given by
gsl_linalg_QR_decomp. The solution is returned in x. The
residual is computed as a by-product and stored in residual.
This function applies the matrix Q^T encoded in the decomposition (QR,tau) to the vector v, storing the result Q^T v in v. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q^T.
This function applies the matrix Q encoded in the decomposition (QR,tau) to the vector v, storing the result Q v in v. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q.
This function applies the matrix Q^T encoded in the decomposition (QR,tau) to the matrix A, storing the result Q^T A in A. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q^T.
This function solves the triangular system R x = b for
x. It may be useful if the product b' = Q^T b has already
been computed using gsl_linalg_QR_QTvec.
This function solves the triangular system R x = b for x
in-place. On input x should contain the right-hand side b
and is replaced by the solution on output. This function may be useful if
the product b' = Q^T b has already been computed using
gsl_linalg_QR_QTvec.
This function unpacks the encoded QR decomposition (QR,tau) into the matrices Q and R, where Q is M-by-M and R is M-by-N.
This function solves the system R x = Q^T b for x. It can be used when the QR decomposition of a matrix is available in unpacked form as (Q, R).
This function performs a rank-1 update w v^T of the QR decomposition (Q, R). The update is given by Q'R' = Q (R + w v^T) where the output matrices Q' and R' are also orthogonal and right triangular. Note that w is destroyed by the update.
This function solves the triangular system R x = b for the N-by-N matrix R.
This function solves the triangular system R x = b in-place. On input x should contain the right-hand side b, which is replaced by the solution on output.
Next: QR Decomposition with Column Pivoting, Previous: LU Decomposition, Up: Linear Algebra [Index]
Next: Spherical Vector Distributions, Previous: The Logistic Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the Pareto distribution of order a. The distribution function is,
p(x) dx = (a/b) / (x/b)^{a+1} dx
for x >= b.
This function computes the probability density p(x) at x for a Pareto distribution with exponent a and scale b, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Pareto distribution with exponent a and scale b.
Next: Copying matrices, Previous: Matrix views, Up: Matrices [Index]
In general there are two ways to access an object, by reference or by copying. The functions described in this section create vector views which allow access to a row or column of a matrix by reference. Modifying elements of the view is equivalent to modifying the matrix, since both the vector view and the matrix point to the same memory block.
These functions return a vector view of the i-th row of the matrix
m. The data pointer of the new vector is set to null if
i is out of range.
The function gsl_vector_const_row is equivalent to
gsl_matrix_row but can be used for matrices which are declared
const.
These functions return a vector view of the j-th column of the
matrix m. The data pointer of the new vector is set to
null if j is out of range.
The function gsl_vector_const_column is equivalent to
gsl_matrix_column but can be used for matrices which are declared
const.
These functions return a vector view of the i-th row of the matrix
m beginning at offset elements past the first column and
containing n elements. The data pointer of the new vector
is set to null if i, offset, or n are out of range.
The function gsl_vector_const_subrow is equivalent to
gsl_matrix_subrow but can be used for matrices which are declared
const.
These functions return a vector view of the j-th column of the matrix
m beginning at offset elements past the first row and
containing n elements. The data pointer of the new vector
is set to null if j, offset, or n are out of range.
The function gsl_vector_const_subcolumn is equivalent to
gsl_matrix_subcolumn but can be used for matrices which are declared
const.
These functions return a vector view of the diagonal of the matrix m. The matrix m is not required to be square. For a rectangular matrix the length of the diagonal is the same as the smaller dimension of the matrix.
The function gsl_matrix_const_diagonal is equivalent to
gsl_matrix_diagonal but can be used for matrices which are
declared const.
These functions return a vector view of the k-th subdiagonal of the matrix m. The matrix m is not required to be square. The diagonal of the matrix corresponds to k = 0.
The function gsl_matrix_const_subdiagonal is equivalent to
gsl_matrix_subdiagonal but can be used for matrices which are
declared const.
These functions return a vector view of the k-th superdiagonal of the matrix m. The matrix m is not required to be square. The diagonal of the matrix corresponds to k = 0.
The function gsl_matrix_const_superdiagonal is equivalent to
gsl_matrix_superdiagonal but can be used for matrices which are
declared const.
Next: Copying matrices, Previous: Matrix views, Up: Matrices [Index]
Next: Elliptic Integrals, Previous: Dilogarithm, Up: Special Functions [Index]
The following functions allow for the propagation of errors when combining quantities by multiplication. The functions are declared in the header file gsl_sf_elementary.h.
This function multiplies x and y storing the product and its associated error in result.
This function multiplies x and y with associated absolute errors dx and dy. The product xy +/- xy \sqrt((dx/x)^2 +(dy/y)^2) is stored in result.
Next: Multimin Stopping Criteria, Previous: Providing a function to minimize, Up: Multidimensional Minimization [Index]
The following function drives the iteration of each algorithm. The function performs one iteration to update the state of the minimizer. The same function works for all minimizers so that different methods can be substituted at runtime without modifications to the code.
These functions perform a single iteration of the minimizer s.
If the iteration encounters an unexpected problem then an error code
will be returned. The error code GSL_ENOPROG signifies that
the minimizer is unable to improve on its current estimate, either due
to numerical difficulty or because a genuine local minimum has been
reached.
The minimizer maintains a current best estimate of the minimum at all times. This information can be accessed with the following auxiliary functions,
These functions return the current best estimate of the location of the minimum, the value of the function at that point, its gradient, the last step increment of the estimate, and minimizer specific characteristic size for the minimizer s.
This function resets the minimizer s to use the current point as a new starting point.
Next: 2D Interpolation Functions, Previous: 1D Interpolation References and Further Reading, Up: Interpolation [Index]
Given a set of x coordinates x_1,...,x_m and a set of y coordinates y_1,...,y_n, each in increasing order, plus a set of function values z_{ij} for each grid point (x_i,y_j), the routines described in this section compute a continuous interpolation function z(x,y) such that z(x_i,y_j) = z_{ij}.
gsl-ref-html-2.3/Regular-Modified-Bessel-Functions-_002d-Fractional-Order.html 0000664 0001750 0001750 00000012646 13055414521 025065 0 ustar edd eddNext: Irregular Modified Bessel Functions - Fractional Order, Previous: Irregular Bessel Functions - Fractional Order, Up: Bessel Functions [Index]
These routines compute the regular modified Bessel function of fractional order \nu, I_\nu(x) for x>0, \nu>0.
These routines compute the scaled regular modified Bessel function of fractional order \nu, \exp(-|x|)I_\nu(x) for x>0, \nu>0.
Next: GNU General Public License, Previous: Autoconf Macros, Up: Top [Index]
The prototypes for the low-level CBLAS functions are declared in the file gsl_cblas.h. For the definition of the functions consult the documentation available from Netlib (see BLAS References and Further Reading).
| • Level 1 CBLAS Functions: | ||
| • Level 2 CBLAS Functions: | ||
| • Level 3 CBLAS Functions: | ||
| • GSL CBLAS Examples: |
Previous: Resampling from 2D histograms, Up: Histograms [Index]
This program demonstrates two features of two-dimensional histograms. First a 10-by-10 two-dimensional histogram is created with x and y running from 0 to 1. Then a few sample points are added to the histogram, at (0.3,0.3) with a height of 1, at (0.8,0.1) with a height of 5 and at (0.7,0.9) with a height of 0.5. This histogram with three events is used to generate a random sample of 1000 simulated events, which are printed out.
#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_histogram2d.h>
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
gsl_histogram2d * h = gsl_histogram2d_alloc (10, 10);
gsl_histogram2d_set_ranges_uniform (h,
0.0, 1.0,
0.0, 1.0);
gsl_histogram2d_accumulate (h, 0.3, 0.3, 1);
gsl_histogram2d_accumulate (h, 0.8, 0.1, 5);
gsl_histogram2d_accumulate (h, 0.7, 0.9, 0.5);
gsl_rng_env_setup ();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
{
int i;
gsl_histogram2d_pdf * p
= gsl_histogram2d_pdf_alloc (h->nx, h->ny);
gsl_histogram2d_pdf_init (p, h);
for (i = 0; i < 1000; i++) {
double x, y;
double u = gsl_rng_uniform (r);
double v = gsl_rng_uniform (r);
gsl_histogram2d_pdf_sample (p, u, v, &x, &y);
printf ("%g %g\n", x, y);
}
gsl_histogram2d_pdf_free (p);
}
gsl_histogram2d_free (h);
gsl_rng_free (r);
return 0;
}
Next: Definition of Carlson Forms, Up: Elliptic Integrals [Index]
The Legendre forms of elliptic integrals F(\phi,k), E(\phi,k) and \Pi(\phi,k,n) are defined by,
F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t))) E(\phi,k) = \int_0^\phi dt \sqrt((1 - k^2 \sin^2(t))) Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))
The complete Legendre forms are denoted by K(k) = F(\pi/2, k) and E(k) = E(\pi/2, k).
The notation used here is based on Carlson, Numerische Mathematik 33 (1979) 1 and differs slightly from that used by Abramowitz & Stegun, where the functions are given in terms of the parameter m = k^2 and n is replaced by -n.
gsl-ref-html-2.3/Code-Reuse.html 0000664 0001750 0001750 00000007754 13055414555 014600 0 ustar edd eddPrevious: Deprecated Functions, Up: Using the library [Index]
Where possible the routines in the library have been written to avoid
dependencies between modules and files. This should make it possible to
extract individual functions for use in your own applications, without
needing to have the whole library installed. You may need to define
certain macros such as GSL_ERROR and remove some #include
statements in order to compile the files as standalone units. Reuse of
the library code in this way is encouraged, subject to the terms of the
GNU General Public License.
Next: Mass and Weight, Previous: Printers Units, Up: Physical Constants [Index]
GSL_CONST_MKSA_MICRONThe length of 1 micron.
GSL_CONST_MKSA_HECTAREThe area of 1 hectare.
GSL_CONST_MKSA_ACREThe area of 1 acre.
GSL_CONST_MKSA_LITERThe volume of 1 liter.
GSL_CONST_MKSA_US_GALLONThe volume of 1 US gallon.
GSL_CONST_MKSA_CANADIAN_GALLONThe volume of 1 Canadian gallon.
GSL_CONST_MKSA_UK_GALLONThe volume of 1 UK gallon.
GSL_CONST_MKSA_QUARTThe volume of 1 quart.
GSL_CONST_MKSA_PINTThe volume of 1 pint.
Next: Combination properties, Previous: Combination allocation, Up: Combinations [Index]
The following function can be used to access the elements of a combination.
This function returns the value of the i-th element of the
combination c. If i lies outside the allowed range of 0 to
k-1 then the error handler is invoked and 0 is returned. An inline version of this function is used when HAVE_INLINE is defined.
Next: Riemann Zeta Function Minus One, Up: Zeta Functions [Index]
The Riemann zeta function is defined by the infinite sum \zeta(s) = \sum_{k=1}^\infty k^{-s}.
These routines compute the Riemann zeta function \zeta(n) for integer n, n \ne 1.
These routines compute the Riemann zeta function \zeta(s) for arbitrary s, s \ne 1.
Next: Conversion Functions, Previous: Trigonometric Functions for Complex Arguments, Up: Trigonometric Functions [Index]
These routines compute \log(\sinh(x)) for x > 0.
These routines compute \log(\cosh(x)) for any x.
Previous: GCC warning options for numerical programs, Up: Debugging Numerical Programs [Index]
The following books are essential reading for anyone writing and debugging numerical programs with GCC and GDB.
For a tutorial introduction to the GNU C Compiler and related programs, see
Previous: Sparse Iterative Solvers Types, Up: Sparse Iterative Solvers [Index]
The following functions are provided to allocate storage for the sparse linear solvers and iterate the system to a solution.
This function allocates a workspace for the iterative solution of n-by-n sparse matrix systems. The iterative solver type is specified by T. The argument m specifies the size of the solution candidate subspace {\cal K}_m. The dimension m may be set to 0 in which case a reasonable default value is used.
This function frees the memory associated with the workspace w.
This function returns a string pointer to the name of the solver.
This function performs one iteration of the iterative method for the sparse linear system specfied by the matrix A, right hand side vector b and solution vector x. On input, x must be set to an initial guess for the solution. On output, x is updated to give the current solution estimate. The parameter tol specifies the relative tolerance between the residual norm and norm of b in order to check for convergence. When the following condition is satisfied:
|| A x - b || <= tol * || b ||
the method has converged, the function returns GSL_SUCCESS and
the final solution is provided in x. Otherwise, the function
returns GSL_CONTINUE to signal that more iterations are
required. Here, || \cdot || represents the Euclidean norm.
The input matrix A may be in triplet or compressed column
format.
This function returns the current residual norm
||r|| = ||A x - b||, which is updated after each call to
gsl_splinalg_itersolve_iterate.
Previous: Sparse Iterative Solvers Types, Up: Sparse Iterative Solvers [Index]
Next: MISER, Previous: Monte Carlo Interface, Up: Monte Carlo Integration [Index]
The plain Monte Carlo algorithm samples points randomly from the integration region to estimate the integral and its error. Using this algorithm the estimate of the integral E(f; N) for N randomly distributed points x_i is given by,
E(f; N) = = V <f> = (V / N) \sum_i^N f(x_i)
where V is the volume of the integration region. The error on this estimate \sigma(E;N) is calculated from the estimated variance of the mean,
\sigma^2 (E; N) = (V^2 / N^2) \sum_i^N (f(x_i) - <f>)^2.
For large N this variance decreases asymptotically as \Var(f)/N, where \Var(f) is the true variance of the function over the integration region. The error estimate itself should decrease as \sigma(f)/\sqrt{N}. The familiar law of errors decreasing as 1/\sqrt{N} applies—to reduce the error by a factor of 10 requires a 100-fold increase in the number of sample points.
The functions described in this section are declared in the header file gsl_monte_plain.h.
This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions.
This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.
This routines uses the plain Monte Carlo algorithm to integrate the function f over the dim-dimensional hypercubic region defined by the lower and upper limits in the arrays xl and xu, each of size dim. The integration uses a fixed number of function calls calls, and obtains random sampling points using the random number generator r. A previously allocated workspace s must be supplied. The result of the integration is returned in result, with an estimated absolute error abserr.
This function frees the memory associated with the integrator state s.
Next: MISER, Previous: Monte Carlo Interface, Up: Monte Carlo Integration [Index]
Next: DWT References, Previous: DWT Transform Functions, Up: Wavelet Transforms [Index]
The following program demonstrates the use of the one-dimensional wavelet transform functions. It computes an approximation to an input signal (of length 256) using the 20 largest components of the wavelet transform, while setting the others to zero.
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_sort.h>
#include <gsl/gsl_wavelet.h>
int
main (int argc, char **argv)
{
(void)(argc); /* avoid unused parameter warning */
int i, n = 256, nc = 20;
double *data = malloc (n * sizeof (double));
double *abscoeff = malloc (n * sizeof (double));
size_t *p = malloc (n * sizeof (size_t));
FILE * f;
gsl_wavelet *w;
gsl_wavelet_workspace *work;
w = gsl_wavelet_alloc (gsl_wavelet_daubechies, 4);
work = gsl_wavelet_workspace_alloc (n);
f = fopen (argv[1], "r");
for (i = 0; i < n; i++)
{
fscanf (f, "%lg", &data[i]);
}
fclose (f);
gsl_wavelet_transform_forward (w, data, 1, n, work);
for (i = 0; i < n; i++)
{
abscoeff[i] = fabs (data[i]);
}
gsl_sort_index (p, abscoeff, 1, n);
for (i = 0; (i + nc) < n; i++)
data[p[i]] = 0;
gsl_wavelet_transform_inverse (w, data, 1, n, work);
for (i = 0; i < n; i++)
{
printf ("%g\n", data[i]);
}
gsl_wavelet_free (w);
gsl_wavelet_workspace_free (work);
free (data);
free (abscoeff);
free (p);
return 0;
}
The output can be used with the GNU plotutils graph program,
$ ./a.out ecg.dat > dwt.txt $ graph -T ps -x 0 256 32 -h 0.3 -a dwt.txt > dwt.ps
Next: Evaluation of B-spline basis function derivatives, Previous: Constructing the knots vector, Up: Basis Splines [Index]
This function evaluates all B-spline basis functions at the position
x and stores them in the vector B, so that the i-th element
is B_i(x). The vector B must be of length
n = nbreak + k - 2. This value may also be obtained by calling
gsl_bspline_ncoeffs.
Computing all the basis functions at once is more efficient than
computing them individually, due to the nature of the defining
recurrence relation.
This function evaluates all potentially nonzero B-spline basis functions at the position x and stores them in the vector Bk, so that the i-th element is B_(istart+i)(x). The last element of Bk is B_(iend)(x). The vector Bk must be of length k. By returning only the nonzero basis functions, this function allows quantities involving linear combinations of the B_i(x) to be computed without unnecessary terms (such linear combinations occur, for example, when evaluating an interpolated function).
This function returns the number of B-spline coefficients given by n = nbreak + k - 2.
Next: Initializing the Multidimensional Minimizer, Previous: Multimin Overview, Up: Multidimensional Minimization [Index]
Note that the minimization algorithms can only search for one local minimum at a time. When there are several local minima in the search area, the first minimum to be found will be returned; however it is difficult to predict which of the minima this will be. In most cases, no error will be reported if you try to find a local minimum in an area where there is more than one.
It is also important to note that the minimization algorithms find local minima; there is no way to determine whether a minimum is a global minimum of the function in question.
gsl-ref-html-2.3/QAWC-adaptive-integration-for-Cauchy-principal-values.html 0000664 0001750 0001750 00000012370 13055414454 024752 0 ustar edd eddNext: QAWS adaptive integration for singular functions, Previous: QAGI adaptive integration on infinite intervals, Up: Numerical Integration [Index]
This function computes the Cauchy principal value of the integral of f over (a,b), with a singularity at c,
I = \int_a^b dx f(x) / (x - c)
The adaptive bisection algorithm of QAG is used, with modifications to ensure that subdivisions do not occur at the singular point x = c. When a subinterval contains the point x = c or is close to it then a special 25-point modified Clenshaw-Curtis rule is used to control the singularity. Further away from the singularity the algorithm uses an ordinary 15-point Gauss-Kronrod integration rule.
Next: Running Statistics Current Statistics, Previous: Running Statistics Initializing the Accumulator, Up: Running Statistics [Index]
This function adds the data point x to the statistical accumulator, updating calculations of the mean, variance, standard deviation, skewness, kurtosis, and median.
This function returns the number of data so far added to the accumulator.
Next: Correlation, Previous: Autocorrelation, Up: Statistics [Index]
This function computes the covariance of the datasets data1 and data2 which must both be of the same length n.
covar = (1/(n - 1)) \sum_{i = 1}^{n} (x_i - \Hat x) (y_i - \Hat y)
This function computes the covariance of the datasets data1 and data2 using the given values of the means, mean1 and mean2. This is useful if you have already computed the means of data1 and data2 and want to avoid recomputing them.
Next: Coulomb Wave Function Normalization Constant, Previous: Normalized Hydrogenic Bound States, Up: Coulomb Functions [Index]
The Coulomb wave functions F_L(\eta,x), G_L(\eta,x) are
described in Abramowitz & Stegun, Chapter 14. Because there can be a
large dynamic range of values for these functions, overflows are handled
gracefully. If an overflow occurs, GSL_EOVRFLW is signalled and
exponent(s) are returned through the modifiable parameters exp_F,
exp_G. The full solution can be reconstructed from the following
relations,
F_L(eta,x) = fc[k_L] * exp(exp_F) G_L(eta,x) = gc[k_L] * exp(exp_G) F_L'(eta,x) = fcp[k_L] * exp(exp_F) G_L'(eta,x) = gcp[k_L] * exp(exp_G)
This function computes the Coulomb wave functions F_L(\eta,x),
G_{L-k}(\eta,x) and their derivatives
F'_L(\eta,x),
G'_{L-k}(\eta,x)
with respect to x. The parameters are restricted to L,
L-k > -1/2, x > 0 and integer k. Note that L
itself is not restricted to being an integer. The results are stored in
the parameters F, G for the function values and Fp,
Gp for the derivative values. If an overflow occurs,
GSL_EOVRFLW is returned and scaling exponents are stored in
the modifiable parameters exp_F, exp_G.
This function computes the Coulomb wave function F_L(\eta,x) for L = Lmin \dots Lmin + kmax, storing the results in fc_array. In the case of overflow the exponent is stored in F_exponent.
This function computes the functions F_L(\eta,x), G_L(\eta,x) for L = Lmin \dots Lmin + kmax storing the results in fc_array and gc_array. In the case of overflow the exponents are stored in F_exponent and G_exponent.
This function computes the functions F_L(\eta,x), G_L(\eta,x) and their derivatives F'_L(\eta,x), G'_L(\eta,x) for L = Lmin \dots Lmin + kmax storing the results in fc_array, gc_array, fcp_array and gcp_array. In the case of overflow the exponents are stored in F_exponent and G_exponent.
This function computes the Coulomb wave function divided by the argument F_L(\eta, x)/x for L = Lmin \dots Lmin + kmax, storing the results in fc_array. In the case of overflow the exponent is stored in F_exponent. This function reduces to spherical Bessel functions in the limit \eta \to 0.
Next: Coulomb Wave Function Normalization Constant, Previous: Normalized Hydrogenic Bound States, Up: Coulomb Functions [Index]
Next: Permutations, Previous: Special Functions, Up: Top [Index]
The functions described in this chapter provide a simple vector and matrix interface to ordinary C arrays. The memory management of these arrays is implemented using a single underlying type, known as a block. By writing your functions in terms of vectors and matrices you can pass a single structure containing both data and dimensions as an argument without needing additional function parameters. The structures are compatible with the vector and matrix formats used by BLAS routines.
| • Data types: | ||
| • Blocks: | ||
| • Vectors: | ||
| • Matrices: | ||
| • Vector and Matrix References and Further Reading: |
Next: Synchrotron Functions, Previous: Power Function, Up: Special Functions [Index]
The polygamma functions of order n are defined by
\psi^{(n)}(x) = (d/dx)^n \psi(x) = (d/dx)^{n+1} \log(\Gamma(x))
where \psi(x) = \Gamma'(x)/\Gamma(x) is known as the digamma function. These functions are declared in the header file gsl_sf_psi.h.
| • Digamma Function: | ||
| • Trigamma Function: | ||
| • Polygamma Function: |
Next: Radial Mathieu Functions, Previous: Mathieu Function Characteristic Values, Up: Mathieu Functions [Index]
These routines compute the angular Mathieu functions ce_n(q,x) and se_n(q,x), respectively.
These routines compute a series of the angular Mathieu functions ce_n(q,x) and se_n(q,x) of order n from nmin to nmax inclusive, storing the results in the array result_array.
Next: Function Index, Previous: GNU General Public License, Up: Top [Index]
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Next: Function Index, Previous: GNU General Public License, Up: Top [Index]
Next: Initializing matrix elements, Previous: Matrix allocation, Up: Matrices [Index]
The functions for accessing the elements of a matrix use the same range
checking system as vectors. You can turn off range checking by recompiling
your program with the preprocessor definition
GSL_RANGE_CHECK_OFF.
The elements of the matrix are stored in “C-order”, where the second
index moves continuously through memory. More precisely, the element
accessed by the function gsl_matrix_get(m,i,j) and
gsl_matrix_set(m,i,j,x) is
m->data[i * m->tda + j]
where tda is the physical row-length of the matrix.
This function returns the (i,j)-th element of a matrix
m. If i or j lie outside the allowed range of 0 to
n1-1 and 0 to n2-1 then the error handler is invoked and 0
is returned. An inline version of this function is used when HAVE_INLINE is defined.
This function sets the value of the (i,j)-th element of a
matrix m to x. If i or j lies outside the
allowed range of 0 to n1-1 and 0 to n2-1 then the error
handler is invoked. An inline version of this function is used when HAVE_INLINE is defined.
These functions return a pointer to the (i,j)-th element of a
matrix m. If i or j lie outside the allowed range of
0 to n1-1 and 0 to n2-1 then the error handler is invoked
and a null pointer is returned. Inline versions of these functions are used when HAVE_INLINE is defined.
Next: Initializing matrix elements, Previous: Matrix allocation, Up: Matrices [Index]
Next: The Levy alpha-Stable Distributions, Previous: The Rayleigh Tail Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the Landau distribution. The probability distribution for Landau random variates is defined analytically by the complex integral,
p(x) = (1/(2 \pi i)) \int_{c-i\infty}^{c+i\infty} ds exp(s log(s) + x s)
For numerical purposes it is more convenient to use the following equivalent form of the integral,
p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).
This function computes the probability density p(x) at x for the Landau distribution using an approximation to the formula given above.
Next: Histogram Statistics, Previous: Updating and accessing histogram elements, Up: Histograms [Index]
The following functions are used by the access and update routines to locate the bin which corresponds to a given x coordinate.
This function finds and sets the index i to the bin number which
covers the coordinate x in the histogram h. The bin is
located using a binary search. The search includes an optimization for
histograms with uniform range, and will return the correct bin
immediately in this case. If x is found in the range of the
histogram then the function sets the index i and returns
GSL_SUCCESS. If x lies outside the valid range of the
histogram then the function returns GSL_EDOM and the error
handler is invoked.
Next: Matrix properties, Previous: Matrix operations, Up: Matrices [Index]
The following operations are only defined for real matrices.
This function returns the maximum value in the matrix m.
This function returns the minimum value in the matrix m.
This function returns the minimum and maximum values in the matrix m, storing them in min_out and max_out.
This function returns the indices of the maximum value in the matrix m, storing them in imax and jmax. When there are several equal maximum elements then the first element found is returned, searching in row-major order.
This function returns the indices of the minimum value in the matrix m, storing them in imin and jmin. When there are several equal minimum elements then the first element found is returned, searching in row-major order.
This function returns the indices of the minimum and maximum values in the matrix m, storing them in (imin,jmin) and (imax,jmax). When there are several equal minimum or maximum elements then the first elements found are returned, searching in row-major order.
Next: Sparse Matrices Conversion Between Sparse and Dense, Previous: Sparse Matrices Finding Maximum and Minimum Elements, Up: Sparse Matrices [Index]
GSL supports compressed column storage (CCS) and compressed row storage (CRS) formats.
This function creates a sparse matrix in compressed column format from the input sparse matrix T which must be in triplet format. A pointer to a newly allocated matrix is returned. The calling function should free the newly allocated matrix when it is no longer needed.
This function creates a sparse matrix in compressed row format from the input sparse matrix T which must be in triplet format. A pointer to a newly allocated matrix is returned. The calling function should free the newly allocated matrix when it is no longer needed.
The problem of multidimensional root finding requires the simultaneous solution of n equations, f_i, in n variables, x_i,
f_i (x_1, ..., x_n) = 0 for i = 1 ... n.
In general there are no bracketing methods available for n dimensional systems, and no way of knowing whether any solutions exist. All algorithms proceed from an initial guess using a variant of the Newton iteration,
x -> x' = x - J^{-1} f(x)
where x, f are vector quantities and J is the Jacobian matrix J_{ij} = d f_i / d x_j. Additional strategies can be used to enlarge the region of convergence. These include requiring a decrease in the norm |f| on each step proposed by Newton’s method, or taking steepest-descent steps in the direction of the negative gradient of |f|.
Several root-finding algorithms are available within a single framework. The user provides a high-level driver for the algorithms, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,
The evaluation of the Jacobian matrix can be problematic, either because programming the derivatives is intractable or because computation of the n^2 terms of the matrix becomes too expensive. For these reasons the algorithms provided by the library are divided into two classes according to whether the derivatives are available or not.
The state for solvers with an analytic Jacobian matrix is held in a
gsl_multiroot_fdfsolver struct. The updating procedure requires
both the function and its derivatives to be supplied by the user.
The state for solvers which do not use an analytic Jacobian matrix is
held in a gsl_multiroot_fsolver struct. The updating procedure
uses only function evaluations (not derivatives). The algorithms
estimate the matrix J or J^{-1} by approximate methods.
Next: Approximate Comparison of Floating Point Numbers, Previous: Testing for Odd and Even Numbers, Up: Mathematical Functions [Index]
Note that the following macros perform multiple evaluations of their arguments, so they should not be used with arguments that have side effects (such as a call to a random number generator).
This macro returns the maximum of a and b. It is defined
as ((a) > (b) ? (a):(b)).
This macro returns the minimum of a and b. It is defined as
((a) < (b) ? (a):(b)).
This function returns the maximum of the double precision numbers
a and b using an inline function. The use of a function
allows for type checking of the arguments as an extra safety feature. On
platforms where inline functions are not available the macro
GSL_MAX will be automatically substituted.
This function returns the minimum of the double precision numbers
a and b using an inline function. The use of a function
allows for type checking of the arguments as an extra safety feature. On
platforms where inline functions are not available the macro
GSL_MIN will be automatically substituted.
These functions return the maximum or minimum of the integers a
and b using an inline function. On platforms where inline
functions are not available the macros GSL_MAX or GSL_MIN
will be automatically substituted.
These functions return the maximum or minimum of the long doubles a
and b using an inline function. On platforms where inline
functions are not available the macros GSL_MAX or GSL_MIN
will be automatically substituted.
Next: Approximate Comparison of Floating Point Numbers, Previous: Testing for Odd and Even Numbers, Up: Mathematical Functions [Index]
Next: 9-j Symbols, Previous: 3-j Symbols, Up: Coupling Coefficients [Index]
These routines compute the Wigner 6-j coefficient,
{ja jb jc
jd je jf}
where the arguments are given in half-integer units, ja = two_ja/2, ma = two_ma/2, etc.
Previous: Hurwitz Zeta Function, Up: Zeta Functions [Index]
The eta function is defined by \eta(s) = (1-2^{1-s}) \zeta(s).
These routines compute the eta function \eta(n) for integer n.
These routines compute the eta function \eta(s) for arbitrary s.
Previous: Large Dense Linear Systems Solution Steps, Up: Large Dense Linear Systems [Index]
This function allocates a workspace for solving large linear least squares systems. The least squares matrix X has p columns, but may have any number of rows. The parameter T specifies the method to be used for solving the large least squares system and may be selected from the following choices
This specifies the normal equations approach for solving the least squares system. This method is suitable in cases where performance is critical and it is known that the least squares matrix X is well conditioned. The size of this workspace is O(p^2).
This specifies the sequential Tall Skinny QR (TSQR) approach for solving the least squares system. This method is a good general purpose choice for large systems, but requires about twice as many operations as the normal equations method for n >> p. The size of this workspace is O(p^2).
This function frees the memory associated with the workspace w.
This function returns a string pointer to the name of the multilarge solver.
This function resets the workspace w so it can begin to accumulate a new least squares system.
These functions define a regularization matrix
L = diag(l_0,l_1,...,l_{p-1}).
The diagonal matrix element l_i is provided by the
ith element of the input vector L.
The block (X,y) is converted to standard form and
the parameters (\tilde{X},\tilde{y}) are stored in Xs
and ys on output. Xs and ys have the same dimensions as
X and y. Optional data weights may be supplied in the
vector w. In order to apply this transformation,
L^{-1} must exist and so none of the l_i
may be zero. After the standard form system has been solved,
use gsl_multilarge_linear_genform1 to recover the original solution vector.
It is allowed to have X = Xs and y = ys for an in-place transform.
This function calculates the QR decomposition of the m-by-p regularization matrix
L. L must have m \ge p. On output,
the Householder scalars are stored in the vector tau of size p.
These outputs will be used by gsl_multilarge_linear_wstdform2 to complete the
transformation to standard form.
These functions convert a block of rows (X,y,w) to standard
form (\tilde{X},\tilde{y}) which are stored in Xs and ys
respectively. X, y, and w must all have the same number of rows.
The m-by-p regularization matrix L is specified by the inputs
LQR and Ltau, which are outputs from gsl_multilarge_linear_L_decomp.
Xs and ys have the same dimensions as X and y. After the
standard form system has been solved, use gsl_multilarge_linear_genform2 to
recover the original solution vector. Optional data weights may be supplied in the
vector w, where W = diag(w).
This function accumulates the standard form block (X,y) into the current least squares system. X and y have the same number of rows, which can be arbitrary. X must have p columns. For the TSQR method, X and y are destroyed on output. For the normal equations method, they are both unchanged.
After all blocks (X_i,y_i) have been accumulated into the large least squares system, this function will compute the solution vector which is stored in c on output. The regularization parameter \lambda is provided in lambda. On output, rnorm contains the residual norm ||y - X c||_W and snorm contains the solution norm ||L c||.
After a regularized system has been solved with L = diag(\l_0,\l_1,...,\l_{p-1}), this function backtransforms the standard form solution vector cs to recover the solution vector of the original problem c. The diagonal matrix elements l_i are provided in the vector L. It is allowed to have c = cs for an in-place transform.
After a regularized system has been solved with a regularization matrix L, specified by (LQR,Ltau), this function backtransforms the standard form solution cs to recover the solution vector of the original problem, which is stored in c, of length p.
This function computes the L-curve for a large least squares system after it has been fully accumulated into the workspace work. The output vectors reg_param, rho, and eta must all be the same size, and will contain the regularization parameters \lambda_i, residual norms ||y - X c_i||, and solution norms || L c_i || which compose the L-curve, where c_i is the regularized solution vector corresponding to \lambda_i. The user may determine the number of points on the L-curve by adjusting the size of these input arrays. For the TSQR method, the regularization parameters \lambda_i are estimated from the singular values of the triangular R factor. For the normal equations method, they are estimated from the eigenvalues of the X^T X matrix.
This function computes the reciprocal condition number, stored in rcond, of the least squares matrix after it has been accumulated into the workspace work. For the TSQR algorithm, this is accomplished by calculating the SVD of the R factor, which has the same singular values as the matrix X. For the normal equations method, this is done by computing the eigenvalues of X^T X, which could be inaccurate for ill-conditioned matrices X.
Previous: Large Dense Linear Systems Solution Steps, Up: Large Dense Linear Systems [Index]
Next: Combination References and Further Reading, Previous: Reading and writing combinations, Up: Combinations [Index]
The example program below prints all subsets of the set {0,1,2,3} ordered by size. Subsets of the same size are ordered lexicographically.
#include <stdio.h>
#include <gsl/gsl_combination.h>
int
main (void)
{
gsl_combination * c;
size_t i;
printf ("All subsets of {0,1,2,3} by size:\n") ;
for (i = 0; i <= 4; i++)
{
c = gsl_combination_calloc (4, i);
do
{
printf ("{");
gsl_combination_fprintf (stdout, c, " %u");
printf (" }\n");
}
while (gsl_combination_next (c) == GSL_SUCCESS);
gsl_combination_free (c);
}
return 0;
}
Here is the output from the program,
$ ./a.out
All subsets of {0,1,2,3} by size:
{ }
{ 0 }
{ 1 }
{ 2 }
{ 3 }
{ 0 1 }
{ 0 2 }
{ 0 3 }
{ 1 2 }
{ 1 3 }
{ 2 3 }
{ 0 1 2 }
{ 0 1 3 }
{ 0 2 3 }
{ 1 2 3 }
{ 0 1 2 3 }
All 16 subsets are generated, and the subsets of each size are sorted lexicographically.
gsl-ref-html-2.3/Minimization-Overview.html 0000664 0001750 0001750 00000013266 13055414602 017104 0 ustar edd eddNext: Minimization Caveats, Up: One dimensional Minimization [Index]
The minimization algorithms begin with a bounded region known to contain a minimum. The region is described by a lower bound a and an upper bound b, with an estimate of the location of the minimum x.
The value of the function at x must be less than the value of the function at the ends of the interval,
f(a) > f(x) < f(b)
This condition guarantees that a minimum is contained somewhere within the interval. On each iteration a new point x' is selected using one of the available algorithms. If the new point is a better estimate of the minimum, i.e. where f(x') < f(x), then the current estimate of the minimum x is updated. The new point also allows the size of the bounded interval to be reduced, by choosing the most compact set of points which satisfies the constraint f(a) > f(x) < f(b). The interval is reduced until it encloses the true minimum to a desired tolerance. This provides a best estimate of the location of the minimum and a rigorous error estimate.
Several bracketing algorithms are available within a single framework. The user provides a high-level driver for the algorithm, and the library provides the individual functions necessary for each of the steps. There are three main phases of the iteration. The steps are,
The state for the minimizers is held in a gsl_min_fminimizer
struct. The updating procedure uses only function evaluations (not
derivatives).
Next: Minimization Caveats, Up: One dimensional Minimization [Index]
The functions described here evaluate the polynomial
P(x) = c[0] + c[1] x + c[2] x^2 + \dots + c[len-1] x^{len-1} using
Horner’s method for stability. Inline versions of these functions are used when HAVE_INLINE is defined.
This function evaluates a polynomial with real coefficients for the real variable x.
This function evaluates a polynomial with real coefficients for the complex variable z.
This function evaluates a polynomial with complex coefficients for the complex variable z.
This function evaluates a polynomial and its derivatives storing the results in the array res of size lenres. The output array contains the values of d^k P/d x^k for the specified value of x starting with k = 0.
Next: Astronomy and Astrophysics, Up: Physical Constants [Index]
GSL_CONST_MKSA_SPEED_OF_LIGHTThe speed of light in vacuum, c.
GSL_CONST_MKSA_VACUUM_PERMEABILITYThe permeability of free space, \mu_0. This constant is defined in the MKSA system only.
GSL_CONST_MKSA_VACUUM_PERMITTIVITYThe permittivity of free space, \epsilon_0. This constant is defined in the MKSA system only.
GSL_CONST_MKSA_PLANCKS_CONSTANT_HPlanck’s constant, h.
GSL_CONST_MKSA_PLANCKS_CONSTANT_HBARPlanck’s constant divided by 2\pi, \hbar.
GSL_CONST_NUM_AVOGADROAvogadro’s number, N_a.
GSL_CONST_MKSA_FARADAYThe molar charge of 1 Faraday.
GSL_CONST_MKSA_BOLTZMANNThe Boltzmann constant, k.
GSL_CONST_MKSA_MOLAR_GASThe molar gas constant, R_0.
GSL_CONST_MKSA_STANDARD_GAS_VOLUMEThe standard gas volume, V_0.
GSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANTThe Stefan-Boltzmann radiation constant, \sigma.
GSL_CONST_MKSA_GAUSSThe magnetic field of 1 Gauss.
Previous: Irregular Modified Bessel Functions - Fractional Order, Up: Bessel Functions [Index]
These routines compute the location of the s-th positive zero of the Bessel function J_0(x).
These routines compute the location of the s-th positive zero of the Bessel function J_1(x).
These routines compute the location of the s-th positive zero of the Bessel function J_\nu(x). The current implementation does not support negative values of nu.
Next: Applying Permutations, Previous: Permutation properties, Up: Permutations [Index]
This function reverses the elements of the permutation p.
This function computes the inverse of the permutation p, storing the result in inv.
This function advances the permutation p to the next permutation
in lexicographic order and returns GSL_SUCCESS. If no further
permutations are available it returns GSL_FAILURE and leaves
p unmodified. Starting with the identity permutation and
repeatedly applying this function will iterate through all possible
permutations of a given order.
This function steps backwards from the permutation p to the
previous permutation in lexicographic order, returning
GSL_SUCCESS. If no previous permutation is available it returns
GSL_FAILURE and leaves p unmodified.
Next: Running Statistics Quantiles, Previous: Running Statistics Adding Data to the Accumulator, Up: Running Statistics [Index]
This function returns the minimum value added to the accumulator.
This function returns the maximum value added to the accumulator.
This function returns the mean of all data added to the accumulator, defined as
\Hat\mu = (1/N) \sum x_i
This function returns the variance of all data added to the accumulator, defined as
\Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2
This function returns the standard deviation of all data added to the accumulator, defined as the square root of the variance given above.
This function returns the standard deviation of the mean, defined as
sd_mean = \Hat\sigma / \sqrt{N}
This function returns the root mean square of all data added to the accumulator, defined as
rms = \sqrt ( 1/N \sum x_i^2 )
This function returns the skewness of all data added to the accumulator, defined as
skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3
This function returns the kurtosis of all data added to the accumulator, defined as
kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4) - 3
This function returns an estimate of the median of the data added to the accumulator.
Next: Running Statistics Quantiles, Previous: Running Statistics Adding Data to the Accumulator, Up: Running Statistics [Index]
Next: Sparse Matrices Initializing Elements, Previous: Sparse Matrices Allocation, Up: Sparse Matrices [Index]
This function returns element (i,j) of the matrix m. The matrix may be in triplet or compressed format.
This function sets element (i,j) of the matrix m to the value x. The matrix must be in triplet representation.
This function returns a pointer to the (i,j) element of the matrix m. If the (i,j) element is not explicitly stored in the matrix, a null pointer is returned.
Next: Level 2 CBLAS Functions, Up: GSL CBLAS Library [Index]
Next: Level 2 CBLAS Functions, Up: GSL CBLAS Library [Index]
Next: Minimization Examples, Previous: Minimization Stopping Parameters, Up: One dimensional Minimization [Index]
The minimization algorithms described in this section require an initial interval which is guaranteed to contain a minimum—if a and b are the endpoints of the interval and x is an estimate of the minimum then f(a) > f(x) < f(b). This ensures that the function has at least one minimum somewhere in the interval. If a valid initial interval is used then these algorithm cannot fail, provided the function is well-behaved.
The golden section algorithm is the simplest method of bracketing the minimum of a function. It is the slowest algorithm provided by the library, with linear convergence.
On each iteration, the algorithm first compares the subintervals from the endpoints to the current minimum. The larger subinterval is divided in a golden section (using the famous ratio (3-\sqrt 5)/2 = 0.3819660…) and the value of the function at this new point is calculated. The new value is used with the constraint f(a') > f(x') < f(b') to a select new interval containing the minimum, by discarding the least useful point. This procedure can be continued indefinitely until the interval is sufficiently small. Choosing the golden section as the bisection ratio can be shown to provide the fastest convergence for this type of algorithm.
The Brent minimization algorithm combines a parabolic interpolation with the golden section algorithm. This produces a fast algorithm which is still robust.
The outline of the algorithm can be summarized as follows: on each iteration Brent’s method approximates the function using an interpolating parabola through three existing points. The minimum of the parabola is taken as a guess for the minimum. If it lies within the bounds of the current interval then the interpolating point is accepted, and used to generate a smaller interval. If the interpolating point is not accepted then the algorithm falls back to an ordinary golden section step. The full details of Brent’s method include some additional checks to improve convergence.
This is a variant of Brent’s algorithm which uses the safeguarded step-length algorithm of Gill and Murray.
Next: Minimization Examples, Previous: Minimization Stopping Parameters, Up: One dimensional Minimization [Index]
Next: Creating ntuples, Up: N-tuples [Index]
Ntuples are manipulated using the gsl_ntuple struct. This struct
contains information on the file where the ntuple data is stored, a
pointer to the current ntuple data row and the size of the user-defined
ntuple data struct.
typedef struct {
FILE * file;
void * ntuple_data;
size_t size;
} gsl_ntuple;
Previous: Roots of Polynomials Examples, Up: Polynomials [Index]
The balanced-QR method and its error analysis are described in the following papers,
The formulas for divided differences are given in the following texts,
Previous: Real Argument, Up: Dilogarithm [Index]
This function computes the full complex-valued dilogarithm for the complex argument z = r \exp(i \theta). The real and imaginary parts of the result are returned in result_re, result_im.
Next: Debugging Numerical Programs, Previous: Physical Constants, Up: Top [Index]
This chapter describes functions for examining the representation of floating point numbers and controlling the floating point environment of your program. The functions described in this chapter are declared in the header file gsl_ieee_utils.h.
| • Representation of floating point numbers: | ||
| • Setting up your IEEE environment: | ||
| • IEEE References and Further Reading: |
Next: Chebyshev Approximations, Previous: Interpolation, Up: Top [Index]
The functions described in this chapter compute numerical derivatives by finite differencing. An adaptive algorithm is used to find the best choice of finite difference and to estimate the error in the derivative. These functions are declared in the header file gsl_deriv.h.
| • Numerical Differentiation functions: | ||
| • Numerical Differentiation Examples: | ||
| • Numerical Differentiation References: |
Next: Multimin Examples, Previous: Multimin Algorithms with Derivatives, Up: Multidimensional Minimization [Index]
The algorithms described in this section use only the value of the function at each evaluation point.
These methods use the Simplex algorithm of Nelder and Mead. Starting from the initial vector x = p_0, the algorithm constructs an additional n vectors p_i using the step size vector s = step_size as follows:
p_0 = (x_0, x_1, ... , x_n) p_1 = (x_0 + s_0, x_1, ... , x_n) p_2 = (x_0, x_1 + s_1, ... , x_n) ... = ... p_n = (x_0, x_1, ... , x_n + s_n)
These vectors form the n+1 vertices of a simplex in n dimensions. On each iteration the algorithm uses simple geometrical transformations to update the vector corresponding to the highest function value. The geometric transformations are reflection, reflection followed by expansion, contraction and multiple contraction. Using these transformations the simplex moves through the space towards the minimum, where it contracts itself.
After each iteration, the best vertex is returned. Note, that due to the nature of the algorithm not every step improves the current best parameter vector. Usually several iterations are required.
The minimizer-specific characteristic size is calculated as the
average distance from the geometrical center of the simplex to all its
vertices. This size can be used as a stopping criteria, as the
simplex contracts itself near the minimum. The size is returned by the
function gsl_multimin_fminimizer_size.
The nmsimplex2 version of this minimiser is a new O(N) operations
implementation of the earlier O(N^2) operations nmsimplex
minimiser. It uses the same underlying algorithm, but the simplex
updates are computed more efficiently for high-dimensional problems.
In addition, the size of simplex is calculated as the RMS
distance of each vertex from the center rather than the mean distance,
allowing a linear update of this quantity on each step. The memory usage is
O(N^2) for both algorithms.
This method is a variant of nmsimplex2 which initialises the
simplex around the starting point x using a randomly-oriented
set of basis vectors instead of the fixed coordinate axes. The
final dimensions of the simplex are scaled along the coordinate axes by the
vector step_size. The randomisation uses a simple deterministic
generator so that repeated calls to gsl_multimin_fminimizer_set for
a given solver object will vary the orientation in a well-defined way.
Next: Multimin Examples, Previous: Multimin Algorithms with Derivatives, Up: Multidimensional Minimization [Index]
Next: Physical Constant References and Further Reading, Previous: Prefixes, Up: Physical Constants [Index]
The following program demonstrates the use of the physical constants in a calculation. In this case, the goal is to calculate the range of light-travel times from Earth to Mars.
The required data is the average distance of each planet from the Sun in astronomical units (the eccentricities and inclinations of the orbits will be neglected for the purposes of this calculation). The average radius of the orbit of Mars is 1.52 astronomical units, and for the orbit of Earth it is 1 astronomical unit (by definition). These values are combined with the MKSA values of the constants for the speed of light and the length of an astronomical unit to produce a result for the shortest and longest light-travel times in seconds. The figures are converted into minutes before being displayed.
#include <stdio.h>
#include <gsl/gsl_const_mksa.h>
int
main (void)
{
double c = GSL_CONST_MKSA_SPEED_OF_LIGHT;
double au = GSL_CONST_MKSA_ASTRONOMICAL_UNIT;
double minutes = GSL_CONST_MKSA_MINUTE;
/* distance stored in meters */
double r_earth = 1.00 * au;
double r_mars = 1.52 * au;
double t_min, t_max;
t_min = (r_mars - r_earth) / c;
t_max = (r_mars + r_earth) / c;
printf ("light travel time from Earth to Mars:\n");
printf ("minimum = %.1f minutes\n", t_min / minutes);
printf ("maximum = %.1f minutes\n", t_max / minutes);
return 0;
}
Here is the output from the program,
light travel time from Earth to Mars: minimum = 4.3 minutes maximum = 21.0 minutes
Next: Search Stopping Parameters, Previous: Search Bounds and Guesses, Up: One dimensional Root-Finding [Index]
The following functions drive the iteration of each algorithm. Each function performs one iteration to update the state of any solver of the corresponding type. The same functions work for all solvers so that different methods can be substituted at runtime without modifications to the code.
These functions perform a single iteration of the solver s. If the iteration encounters an unexpected problem then an error code will be returned,
GSL_EBADFUNCthe iteration encountered a singular point where the function or its
derivative evaluated to Inf or NaN.
GSL_EZERODIVthe derivative of the function vanished at the iteration point, preventing the algorithm from continuing without a division by zero.
The solver maintains a current best estimate of the root at all times. The bracketing solvers also keep track of the current best interval bounding the root. This information can be accessed with the following auxiliary functions,