Provided by: libmath-gsl-perl_0.39-1build2_amd64 
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 offcial 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) 2008-2011 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.