oracular (3) Math::GSL::Eigen.3pm.gz
NAME
Math::GSL::Eigen - Functions for computing eigenvalues and eigenvectors of matrices
SYNOPSIS
use Math::GSL::Eigen qw/:all/;
DESCRIPTION
Here is a list of all the functions included in this module :
gsl_eigen_symm_alloc($n) - This function returns a workspace for computing eigenvalues of n-by-n real
symmetric matrices.
gsl_eigen_symm_free($w) - This function frees the memory associated with the workspace $w.
gsl_eigen_symm($A, $eval, $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 $eval and are unordered.
gsl_eigen_symmv_alloc($n) - This function returns a workspace for computing eigenvalues and eigenvectors
of n-by-n real symmetric matrices.
gsl_eigen_symmv_free($w) - This function frees the memory associated with the workspace $w.
gsl_eigen_symmv($A, $eval, $evec, $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 $evec.
gsl_eigen_herm_alloc($n) - This function returns a workspace for computing eigenvalues of n-by-n complex
hermitian matrices.
gsl_eigen_herm_free($w) - This function frees the memory associated with the workspace $w.
gsl_eigen_herm($A, $eval, $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 $eval and are unordered.
gsl_eigen_hermv_alloc($n) - This function returns a workspace for computing eigenvalues and eigenvectors
of n-by-n complex hermitian matrices.
gsl_eigen_hermv_free($w) - This function frees the memory associated with the workspace $w.
gsl_eigen_hermv($A, $eval, $evec, $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 $evec.
gsl_eigen_francis_alloc($n) -
gsl_eigen_francis_free($w) - This function frees the memory associated with the workspace $w.
gsl_eigen_francis_T
gsl_eigen_francis
gsl_eigen_francis_Z
gsl_eigen_nonsymm_alloc($n) - This function returns a workspace for computing eigenvalues of n-by-n real
nonsymmetric matrices.
gsl_eigen_nonsymm_free($w) - This function frees the memory associated with the workspace $w.
gsl_eigen_nonsymm_params($compute_t, $balance, $w) - This function sets some parameters which determine
how the eigenvalue problem is solved in subsequent calls to gsl_eigen_nonsymm. If $compute_t is set to 1,
the full Schur form T will be computed by gsl_eigen_nonsymm. If it is set to 0, T will not be computed
(this is the default setting). If balance is set to 1, a balancing transformation is applied to the
matrix prior to computing eigenvalues. This transformation is designed to make the rows and columns of
the matrix have comparable norms, and can result in more accurate eigenvalues for matrices whose entries
vary widely in magnitude.
gsl_eigen_nonsymm($A, $eval, $w) - 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 (1) and the number of converged
eigenvalues is stored in $w->{n_evals}. The converged eigenvalues are stored in the beginning of $eval.
gsl_eigen_nonsymm_Z($A, $eval, $Z, $w) - This function is identical to gsl_eigen_nonsymm except it also
computes the Schur vectors and stores them into the $Z matrix.
gsl_eigen_nonsymmv_alloc($n) - This function allocates a workspace for computing eigenvalues and
eigenvectors of n-by-n real nonsymmetric matrices.
gsl_eigen_nonsymmv_free($w) - This function frees the memory associated with the workspace $w.
gsl_eigen_nonsymmv($A, $eval, $evec, $w) - 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 (1). $eval is a complex vector and $evec is a complex matrix.
gsl_eigen_nonsymmv_Z($A, $eval, $evec, $Z, $w) - This function is identical to gsl_eigen_nonsymmv except
it also saves the Schur vectors into the $Z matrix.
gsl_eigen_gensymm_alloc($n) - This function allocates a workspace for computing eigenvalues of n-by-n
real generalized symmetric-definite eigensystems.
gsl_eigen_gensymm_free($w) - This function frees the memory associated with the workspace $w.
gsl_eigen_gensymm($A, $B, $eval, $w) - This function computes the eigenvalues of the real generalized
symmetric-definite matrix pair ($A, $B), and stores them in the vector $eval. On output, $B contains its
Cholesky decomposition and $A is destroyed.
gsl_eigen_gensymm_standardize
gsl_eigen_gensymmv_alloc($n) - This function allocates a workspace for computing eigenvalues and
eigenvectors of n-by-n real generalized symmetric-definite eigensystems.
gsl_eigen_gensymmv_free($w) - This function frees the memory associated with the workspace $w.
gsl_eigen_gensymmv($A, $B, $eval, $evec, $w) - This function computes the eigenvalues and eigenvectors of
the real generalized symmetric-definite matrix pair ($A, $B), and stores them in $eval vector and $evec
matrix respectively. The computed eigenvectors are normalized to have unit magnitude. On output, $B
contains its Cholesky decomposition and A is destroyed.
gsl_eigen_genherm_alloc($n) - This function allocates a workspace for computing eigenvalues of n-by-n
complex generalized hermitian-definite eigensystems.
gsl_eigen_genherm_free($w) - This function frees the memory associated with the workspace $w.
gsl_eigen_genherm($A, $B, $eval, $w) - This function computes the eigenvalues of the complex generalized
hermitian-definite matrix pair ($A, $B), and stores them in the $eval vector. On output, $B contains its
Cholesky decomposition and $A is destroyed.
gsl_eigen_genherm_standardize
gsl_eigen_genhermv_alloc($n) - This function allocates a workspace for computing eigenvalues and
eigenvectors of n-by-n complex generalized hermitian-definite eigensystems.
gsl_eigen_genhermv_free($w) - This function frees the memory associated with the workspace $w.
gsl_eigen_genhermv($A, $B, $eval, $evec, $w) - This function computes the eigenvalues and eigenvectors of
the complex generalized hermitian-definite matrix pair ($A, $B), and stores them in $eval vector and
$evec matrix respectively. The computed eigenvectors are normalized to have unit magnitude. On output, $B
contains its Cholesky decomposition and $A is destroyed.
gsl_eigen_gen_alloc($n) - This function allocates a workspace for computing eigenvalues of n-by-n real
generalized nonsymmetric eigensystems.
gsl_eigen_gen_free($w) - This function frees the memory associated with the workspace $w.
gsl_eigen_gen_params($compute_s, $compute_t, $balance, $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 $balance parameter is currently ignored, since generalized balancing is not yet implemented.
gsl_eigen_gen($A, $B, $alpha, $beta, $w) - This function computes the eigenvalues of the real generalized
nonsymmetric matrix pair ($A, $B), and stores them as pairs in ($alpha, $beta), where $alpha is complex
and $beta is real, both are vectors. 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, $beta) follows the ordering of the diagonal blocks in the Schur forms S and T. In
rare cases, this function may fail to find all eigenvalues. If this occurs, an error code is returned
(1).
gsl_eigen_gen_QZ($A, $B, $alpha, $beta, $Q, $Z, $w) - This function is identical to gsl_eigen_gen except
it also computes the left and right Schur vectors and stores them into $Q matrix and $Z matrix
respectively.
gsl_eigen_genv_alloc($n) - This function allocates a workspace for computing eigenvalues and eigenvectors
of n-by-n real generalized nonsymmetric eigensystems.
gsl_eigen_genv_free($w) - This function frees the memory associated with the workspace $w.
gsl_eigen_genv($A, $B, $alpha, $beta, $evec, $w) - This function computes eigenvalues and right
eigenvectors of the n-by-n real generalized nonsymmetric matrix pair ($A, $B). The eigenvalues are stored
in ($alpha, $beta) where $alpha is a complex vector and $beta a real vector and the eigenvectors are
stored in $evec complex matrix. It first calls gsl_eigen_gen to compute the eigenvalues, Schur forms, and
Schur vectors. Then it finds eigenvectors of the Schur forms and backtransforms them using the Schur
vectors. The Schur vectors are destroyed in the process, but can be saved by using gsl_eigen_genv_QZ. The
computed eigenvectors are normalized to have unit magnitude. On output, ($A, $B) contains the generalized
Schur form (S, T). If gsl_eigen_gen fails, no eigenvectors are computed, and an error code is returned
(1).
gsl_eigen_genv_QZ($A, $B, $alpha, $beta, $evec, $Q, $Z, $w) - This function is identical to
gsl_eigen_genv except it also computes the left and right Schur vectors and stores them into $Q and $Z
matrices respectively.
gsl_eigen_symmv_sort($eval, $evec, $sort_type) - This function simultaneously sorts the eigenvalues
stored in the vector $eval and the corresponding real eigenvectors stored in the columns of the matrix
$evec according to the value of the parameter $sort_type which is one of the constant included in this
module.
gsl_eigen_hermv_sort($eval, $evec, $sort_type) - This function simultaneously sorts the eigenvalues
stored in the vector $eval and the corresponding real eigenvectors stored in the columns of the matrix
$evec according to the value of the parameter $sort_type which is one of the constant included in this
module.
gsl_eigen_nonsymmv_sort($eval, $evec, $sort_type) - This function simultaneously sorts the eigenvalues
stored in the vector $eval and the corresponding complex eigenvectors stored in the columns of the
complex matrix $evec into ascending or descending order according to the value of the parameter
$sort_type. Only $GSL_EIGEN_SORT_ABS_ASC and $GSL_EIGEN_SORT_ABS_DESC are supported due to the
eigenvalues being complex.
gsl_eigen_gensymmv_sort($eval, $evec, $sort_type) - This function simultaneously sorts the eigenvalues
stored in the vector $eval and the corresponding real eigenvectors stored in the columns of the matrix
$evec according to the value of the parameter $sort_type which is one of the constant included in this
module.
gsl_eigen_genhermv_sort($eval, $evec, $sort_type) - This function simultaneously sorts the eigenvalues
stored in the vector $eval and the corresponding real eigenvectors stored in the columns of the matrix
$evec according to the value of the parameter $sort_type which is one of the constant included in this
module.
gsl_eigen_genv_sort($eval, $evec, $sort_type) - This function simultaneously sorts the eigenvalues stored
in the vector $eval and the corresponding complex eigenvectors stored in the columns of the complex
matrix $evec into ascending or descending order according to the value of the parameter $sort_type. Only
$GSL_EIGEN_SORT_ABS_ASC and $GSL_EIGEN_SORT_ABS_DESC are supported due to the eigenvalues being complex.
gsl_schur_gen_eigvals
gsl_schur_solve_equation
gsl_schur_solve_equation_z
gsl_eigen_jacobi
gsl_eigen_invert_jacobi
This module also includes these constants :
$GSL_EIGEN_SORT_VAL_ASC - ascending order in numerical value
$GSL_EIGEN_SORT_VAL_DESC - descending order in numerical value
$GSL_EIGEN_SORT_ABS_ASC - ascending order in magnitude
$GSL_EIGEN_SORT_ABS_DESC - descending order in magnitude
For more information on the functions, we refer you to the GSL official documentation:
<http://www.gnu.org/software/gsl/manual/html_node/>
EXAMPLES
This example shows how to use the gsl_eigen_symmv functions to find the eigenvalues and eigenvectors of a
matrix.
use Math::GSL::Vector qw/:all/;
use Math::GSL::Matrix qw/:all/;
use Math::GSL::Eigen qw/:all/;
my $w = gsl_eigen_symmv_alloc(2);
my $m = gsl_matrix_alloc(2,2);
gsl_matrix_set($m, 0, 0, 2);
gsl_matrix_set($m, 0, 1, 1);
gsl_matrix_set($m, 1, 0, 1);
gsl_matrix_set($m, 1, 1, 2);
my $eval = gsl_vector_alloc(2);
my $evec = gsl_matrix_alloc(2,2);
gsl_eigen_symmv($m, $eval, $evec, $w);
gsl_eigen_gensymmv_sort($eval, $evec, $GSL_EIGEN_SORT_ABS_ASC);
print "The first eigenvalue is : " . gsl_vector_get($eval, 0) . "\n";
print "The second eigenvalue is : " . gsl_vector_get($eval, 1) . "\n";
my $x = gsl_matrix_get($evec, 0, 0);
my $y = gsl_matrix_get($evec, 0, 1);
print "The first eigenvector is [$x, $y] \n";
$x = gsl_matrix_get($evec, 1, 0);
$y = gsl_matrix_get($evec, 1, 1);
print "The second eigenvector is [$x, $y] \n";
AUTHORS
Jonathan "Duke" Leto <jonathan@leto.net> and Thierry Moisan <thierry.moisan@gmail.com>
COPYRIGHT AND LICENSE
Copyright (C) 2008-2024 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.