Provided by: libmath-gsl-perl_0.39-1build2_amd64 
NAME
Math::GSL::Linalg - Functions for solving linear systems
SYNOPSIS
use Math::GSL::Linalg qw/:all/;
DESCRIPTION
Here is a list of all the functions included in this module :
gsl_linalg_matmult
gsl_linalg_matmult_mod
gsl_linalg_exponential_ss
gsl_linalg_householder_transform
gsl_linalg_complex_householder_transform
gsl_linalg_householder_hm($tau, $v, $A) - 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 $A. The function returns 0 if it succeeded, 1 otherwise.
gsl_linalg_householder_mh($tau, $v, $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 $A.
gsl_linalg_householder_hv($tau, $v, $w) - 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 $w.
gsl_linalg_householder_hm1
gsl_linalg_givens($a,$b,$c,$s)
Performs a Givens rotation on the vector ($a,$b) and stores the answer in $c and $s.
gsl_linalg_givens_gv($v, $i,$j, $c, $s)
Performs a Givens rotation on the $i and $j-th elements of $v, storing them in $c and $s.
gsl_linalg_complex_householder_hm($tau, $v, $A) - Does the same operation than gsl_linalg_householder_hm
but with the complex matrix $A, the complex value $tau and the complex vector $v.
gsl_linalg_complex_householder_mh($tau, $v, $A) - Does the same operation than gsl_linalg_householder_mh
but with the complex matrix $A, the complex value $tau and the complex vector $v.
gsl_linalg_complex_householder_hv($tau, $v, $w) - Does the same operation than gsl_linalg_householder_hv
but with the complex value $tau and the complex vectors $v and $w.
gsl_linalg_hessenberg_decomp($A, $tau) - 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
$tau. tau must be of length N. The function returns 0 if it succeeded, 1 otherwise.
gsl_linalg_hessenberg_unpack($H, $tau, $U) - This function constructs the orthogonal matrix $U from the
information stored in the Hessenberg matrix $H along with the vector $tau. $H and $tau are outputs from
gsl_linalg_hessenberg_decomp.
gsl_linalg_hessenberg_unpack_accum($H, $tau, $V) - 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 $V must have N columns but may have any
number of rows.
gsl_linalg_hessenberg_set_zero($H) - This function sets the lower triangular portion of $H, below the
subdiagonal, to zero. It is useful for clearing out the Householder vectors after calling
gsl_linalg_hessenberg_decomp.
gsl_linalg_hessenberg_submatrix
gsl_linalg_hessenberg
gsl_linalg_hesstri_decomp($A, $B, $U, $V, $work) - This function computes the Hessenberg-Triangular
decomposition of the matrix pair ($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 the vector $work.
gsl_linalg_SV_decomp($A, $V, $S, $work) - 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 vector $work. This routine uses the Golub-Reinsch SVD algorithm.
gsl_linalg_SV_decomp_mod($A, $X, $V, $S, $work) - 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. $A and $V are matrices while $S is a vector.
gsl_linalg_SV_decomp_jacobi($A, $V, $S) - 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. $V is a matrix while $S is a vector.
gsl_linalg_SV_solve($U, $V, $S, $b, $x) - This function solves the system A x = b using the singular
value decomposition ($U, $S, $V) of A given by gsl_linalg_SV_decomp. Only non-zero singular values are
used in computing the solution. The parts of the solution corresponding to singular values of zero are
ignored. Other singular values can be edited out by setting them to zero before calling this function. In
the over-determined case where A has more rows than columns the system is solved in the least squares
sense, returning the solution x which minimizes ||A x - b||_2.
gsl_linalg_LU_decomp($a, $p) - factorize the matrix $a into the LU decomposition PA = LU. On output the
diagonal and upper triangular part of the input matrix A contain the matrix U. The lower triangular part
of the input matrix (excluding the diagonal) contains L. The diagonal elements of L are unity, and are
not stored. The function returns two value, the first is 0 if the operation succeeded, 1 otherwise, and
the second is the sign of the permutation.
gsl_linalg_LU_solve($LU, $p, $b, $x) - This function solves the square system A x = b using the LU
decomposition of the matrix A into (LU, p) given by gsl_linalg_LU_decomp. $LU is a matrix, $p a
permutation and $b and $x are vectors. The function returns 1 if the operation succeeded, 0 otherwise.
gsl_linalg_LU_svx($LU, $p, $x) - This function solves the square system A x = b in-place using the 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. $LU is a matrix, $p a permutation and $x is a vector. The function returns 1 if
the operation succeeded, 0 otherwise.
gsl_linalg_LU_refine($A, $LU, $p, $b, $x, $residual) - This function apply an iterative improvement to
$x, the solution of $A $x = $b, using the LU decomposition of $A into ($LU,$p). The initial residual $r =
$A $x - $b (where $x and $b are vectors) is also computed and stored in the vector $residual.
gsl_linalg_LU_invert($LU, $p, $inverse) - This function computes the inverse of a matrix A from its LU
decomposition stored in the matrix $LU and the permutation $p, storing the result in the matrix $inverse.
gsl_linalg_LU_det($LU, $signum) - This function returns the determinant of a matrix A from its LU
decomposition stored in the $LU matrix. It needs the integer $signum which is the sign of the permutation
returned by gsl_linalg_LU_decomp.
gsl_linalg_LU_lndet($LU) - This function returns the logarithm of the absolute value of the determinant
of a matrix A, from its LU decomposition stored in the $LU matrix.
gsl_linalg_LU_sgndet($LU, $signum) - This functions computes the sign or phase factor of the determinant
of a matrix A, det(A)/|det(A)|, from its LU decomposition, $LU.
gsl_linalg_complex_LU_decomp($A, $p) - Does the same operation than gsl_linalg_LU_decomp but on the
complex matrix $A.
gsl_linalg_complex_LU_solve($LU, $p, $b, $x) - This functions solve the square system A x = b using the
LU decomposition of A into ($LU, $p) given by gsl_linalg_complex_LU_decomp.
gsl_linalg_complex_LU_svx($LU, $p, $x) - Does the same operation than gsl_linalg_LU_svx but on the
complex matrix $LU and the complex vector $x.
gsl_linalg_complex_LU_refine($A, $LU, $p, $b, $x, $residual) - Does the same operation than
gsl_linalg_LU_refine but on the complex matrices $A and $LU and with the complex vectors $b, $x and
$residual.
gsl_linalg_complex_LU_invert($LU, $p, $inverse) - Does the same operation than gsl_linalg_LU_invert but
on the complex matrces $LU and $inverse.
gsl_linalg_complex_LU_det($LU, $signum) - Does the same operation than gsl_linalg_LU_det but on the
complex matrix $LU.
gsl_linalg_complex_LU_lndet($LU) - Does the same operation than gsl_linalg_LU_det but on the complex
matrix $LU.
gsl_linalg_complex_LU_sgndet($LU, $signum) - Does the same operation than gsl_linalg_LU_sgndet but on the
complex matrix $LU.
gsl_linalg_QR_decomp($a, $tau) - factorize 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 $a contain the matrix R. The vector
$tau and the columns of the lower triangular part of the matrix $a contain the Householder coefficients
and Householder vectors which encode the orthogonal matrix Q. The vector tau must be of length k=
min(M,N).
gsl_linalg_QR_solve($QR, $tau, $b, $x) - This function solves the square system A x = b using the QR
decomposition of A into (QR, tau) given by gsl_linalg_QR_decomp. $QR is matrix, and $tau, $b and $x are
vectors.
gsl_linalg_QR_svx($QR, $tau, $x) - This function solves the square system A x = b in-place using the QR
decomposition of A into the matrix $QR and the vector $tau given by gsl_linalg_QR_decomp. On input, the
vector $x should contain the right-hand side b, which is replaced by the solution on output.
gsl_linalg_QR_lssolve($QR, $tau, $b, $x, $residual) - 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 uses the $QR decomposition
of $A into ($QR, $tau) given by gsl_linalg_QR_decomp. The solution is returned in $x. The residual is
computed as a by-product and stored in residual. The function returns 0 if it succeeded, 1 otherwise.
gsl_linalg_QR_QRsolve($Q, $R, $b, $x) - 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, $R). The function
returns 0 if it succeeded, 1 otherwise.
gsl_linalg_QR_Rsolve($QR, $b, $x) - This function solves the triangular system R $x = $b for $x. It may
be useful if the product b' = Q^T b has already been computed using gsl_linalg_QR_QTvec.
gsl_linalg_QR_Rsvx($QR, $x) - This function solves the triangular system R $x = b for $x in-place. On
input $x should contain the right-hand side b and is replaced by the solution on output. This function
may be useful if the product b' = Q^T b has already been computed using gsl_linalg_QR_QTvec. The function
returns 0 if it succeeded, 1 otherwise.
gsl_linalg_QR_update($Q, $R, $b, $x) - This function performs a rank-1 update $w $v**T of the QR
decomposition ($Q, $R). The update is given by Q'R' = Q R + w v^T where the output matrices Q' and R' are
also orthogonal and right triangular. Note that w is destroyed by the update. The function returns 0 if
it succeeded, 1 otherwise.
gsl_linalg_QR_QTvec($QR, $tau, $v) - This function applies the matrix Q^T encoded in the decomposition
($QR,$tau) to the vector $v, storing the result Q^T v in $v. The matrix multiplication is carried out
directly using the encoding of the Householder vectors without needing to form the full matrix Q^T. The
function returns 0 if it succeeded, 1 otherwise.
gsl_linalg_QR_Qvec($QR, $tau, $v) - This function applies the matrix Q encoded in the decomposition
($QR,$tau) to the vector $v, storing the result Q v in $v. The matrix multiplication is carried out
directly using the encoding of the Householder vectors without needing to form the full matrix Q. The
function returns 0 if it succeeded, 1 otherwise.
gsl_linalg_QR_QTmat($QR, $tau, $A) - This function applies the matrix Q^T encoded in the decomposition
($QR,$tau) to the matrix $A, storing the result Q^T A in $A. The matrix multiplication is carried out
directly using the encoding of the Householder vectors without needing to form the full matrix Q^T. The
function returns 0 if it succeeded, 1 otherwise.
gsl_linalg_QR_unpack($QR, $tau, $Q, $R) - This function unpacks the encoded QR decomposition ($QR,$tau)
into the matrices $Q and $R, where $Q is M-by-M and $R is M-by-N. The function returns 0 if it succeeded,
1 otherwise.
gsl_linalg_R_solve($R, $b, $x) - This function solves the triangular system $R $x = $b for the N-by-N
matrix $R. The function returns 0 if it succeeded, 1 otherwise.
gsl_linalg_R_svx($R, $x) - This function solves the triangular system $R $x = b in-place. On input $x
should contain the right-hand side b, which is replaced by the solution on output. The function returns 0
if it succeeded, 1 otherwise.
gsl_linalg_QRPT_decomp($A, $tau, $p, $norm) - 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. There's two value
returned by this function : the first is 0 if the operation succeeded, 1 otherwise. The second is sign of
the permutation. 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 $A contain the Householder
coefficients and vectors which encode the orthogonal matrix Q. The vector tau must be of length
k=\min(M,N). The matrix Q is related to these components by, Q = Q_k ... Q_2 Q_1 where Q_i = I - \tau_i
v_i v_i^T and v_i is the Householder vector v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)). This is the
same storage scheme as used by lapack. The vector norm is a workspace of length N used for column
pivoting. The algorithm used to perform the decomposition is Householder QR with column pivoting (Golub &
Van Loan, Matrix Computations, Algorithm 5.4.1).
gsl_linalg_QRPT_decomp2($A, $q, $r, $tau, $p, $norm) - 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 $r. For the rest, it operates exactly like gsl_linalg_QRPT_decomp
gsl_linalg_QRPT_solve
gsl_linalg_QRPT_svx
gsl_linalg_QRPT_QRsolve
gsl_linalg_QRPT_Rsolve
gsl_linalg_QRPT_Rsvx
gsl_linalg_QRPT_update
gsl_linalg_LQ_decomp
gsl_linalg_LQ_solve_T
gsl_linalg_LQ_svx_T
gsl_linalg_LQ_lssolve_T
gsl_linalg_LQ_Lsolve_T
gsl_linalg_LQ_Lsvx_T
gsl_linalg_L_solve_T
gsl_linalg_LQ_vecQ
gsl_linalg_LQ_vecQT
gsl_linalg_LQ_unpack
gsl_linalg_LQ_update
gsl_linalg_LQ_LQsolve
gsl_linalg_PTLQ_decomp
gsl_linalg_PTLQ_decomp2
gsl_linalg_PTLQ_solve_T
gsl_linalg_PTLQ_svx_T
gsl_linalg_PTLQ_LQsolve_T
gsl_linalg_PTLQ_Lsolve_T
gsl_linalg_PTLQ_Lsvx_T
gsl_linalg_PTLQ_update
gsl_linalg_cholesky_decomp($A) - Factorize the symmetric, positive-definite square matrix $A into the
Cholesky decomposition A = L L^T and stores it into the matrix $A. The funtcion returns 0 if the
operation succeeded, 0 otherwise.
gsl_linalg_cholesky_solve($cholesky, $b, $x) - This function solves the system A x = b using the Cholesky
decomposition of A into the matrix $cholesky given by gsl_linalg_cholesky_decomp. $b and $x are vectors.
The funtcion returns 0 if the operation succeeded, 0 otherwise.
gsl_linalg_cholesky_svx($cholesky, $x) - This function solves the system A x = b in-place using the
Cholesky decomposition of A into the matrix $cholesky given by gsl_linalg_cholesky_decomp. On input the
vector $x should contain the right-hand side b, which is replaced by the solution on output. The funtcion
returns 0 if the operation succeeded, 0 otherwise.
gsl_linalg_cholesky_decomp_unit
gsl_linalg_complex_cholesky_decomp($A) - Factorize the symmetric, positive-definite square matrix $A
which contains complex numbers into the Cholesky decomposition A = L L^T and stores it into the matrix
$A. The funtcion returns 0 if the operation succeeded, 0 otherwise.
gsl_linalg_complex_cholesky_solve($cholesky, $b, $x) - This function solves the system A x = b using the
Cholesky decomposition of A into the matrix $cholesky given by gsl_linalg_complex_cholesky_decomp. $b and
$x are vectors. The funtcion returns 0 if the operation succeeded, 0 otherwise.
gsl_linalg_complex_cholesky_svx($cholesky, $x) - This function solves the system A x = b in-place using
the Cholesky decomposition of A into the matrix $cholesky given by gsl_linalg_complex_cholesky_decomp. On
input the vector $x should contain the right-hand side b, which is replaced by the solution on output.
The funtcion returns 0 if the operation succeeded, 0 otherwise.
gsl_linalg_symmtd_decomp($A, $tau) - 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. $tau is a vector.
gsl_linalg_symmtd_unpack($A, $tau, $Q, $diag, $subdiag) - This function unpacks the encoded symmetric
tridiagonal decomposition ($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 $subdiag.
gsl_linalg_symmtd_unpack_T($A, $diag, $subdiag) - This function unpacks the diagonal and subdiagonal of
the encoded symmetric tridiagonal decomposition ($A, $tau) obtained from gsl_linalg_symmtd_decomp into
the vectors $diag and $subdiag.
gsl_linalg_hermtd_decomp($A, $tau) - 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
orthogonal matrix Q. 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. $A is a complex matrix and $tau a complex vector.
gsl_linalg_hermtd_unpack($A, $tau, $U, $diag, $subdiag) - This function unpacks the encoded tridiagonal
decomposition ($A, $tau) obtained from gsl_linalg_hermtd_decomp into the unitary complex matrix $U, the
real vector of diagonal elements $diag and the real vector of subdiagonal elements $subdiag.
gsl_linalg_hermtd_unpack_T($A, $diag, $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 $subdiag.
gsl_linalg_HH_solve($a, $b, $x) - This function solves the system $A $x = $b directly using Householder
transformations where $A is a matrix, $b and $x vectors. On output the solution is stored in $x and $b is
not modified. $A is destroyed by the Householder transformations.
gsl_linalg_HH_svx($A, $x) - This function solves the system $A $x = b in-place using Householder
transformations where $A is a matrix, $b is a vector. 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.
gsl_linalg_solve_symm_tridiag
gsl_linalg_solve_tridiag
gsl_linalg_solve_symm_cyc_tridiag
gsl_linalg_solve_cyc_tridiag
gsl_linalg_bidiag_decomp
gsl_linalg_bidiag_unpack
gsl_linalg_bidiag_unpack2
gsl_linalg_bidiag_unpack_B
gsl_linalg_balance_matrix
gsl_linalg_balance_accum
gsl_linalg_balance_columns
You have to add the functions you want to use inside the qw /put_funtion_here / with spaces between each function. You can also write use Math::GSL::Complex qw/:all/ to use all available functions of the module.
For more information on the functions, we refer you to the GSL offcial documentation:
<http://www.gnu.org/software/gsl/manual/html_node/>
EXAMPLES
This example shows how to compute the determinant of a matrix with the LU decomposition:
use Math::GSL::Matrix qw/:all/;
use Math::GSL::Permutation qw/:all/;
use Math::GSL::Linalg qw/:all/;
my $Matrix = gsl_matrix_alloc(4,4);
map { gsl_matrix_set($Matrix, 0, $_, $_+1) } (0..3);
gsl_matrix_set($Matrix,1, 0, 2);
gsl_matrix_set($Matrix, 1, 1, 3);
gsl_matrix_set($Matrix, 1, 2, 4);
gsl_matrix_set($Matrix, 1, 3, 1);
gsl_matrix_set($Matrix, 2, 0, 3);
gsl_matrix_set($Matrix, 2, 1, 4);
gsl_matrix_set($Matrix, 2, 2, 1);
gsl_matrix_set($Matrix, 2, 3, 2);
gsl_matrix_set($Matrix, 3, 0, 4);
gsl_matrix_set($Matrix, 3, 1, 1);
gsl_matrix_set($Matrix, 3, 2, 2);
gsl_matrix_set($Matrix, 3, 3, 3);
my $permutation = gsl_permutation_alloc(4);
gsl_permutation_init($permutation);
my ($result, $signum) = gsl_linalg_LU_decomp($Matrix, $permutation);
my $det = gsl_linalg_LU_det($Matrix, $signum);
print "The value of the determinant of the matrix is $det \n";
AUTHORS
Jonathan "Duke" Leto <jonathan@leto.net> and Thierry Moisan <thierry.moisan@gmail.com>
COPYRIGHT AND LICENSE
Copyright (C) 2008-2015 Jonathan "Duke" Leto and Thierry Moisan
This program is free software; you can redistribute it and/or modify it under the same terms as Perl
itself.
perl v5.26.0 2017-08-06 Math::GSL::Linalg(3pm)