

 
Math::GSL::Eigen(3) 
User Contributed Perl Documentation 
Math::GSL::Eigen(3) 
Math::GSL::Eigen  Functions for computing eigenvalues and eigenvectors of
matrices
use Math::GSL::Eigen qw/:all/;
Here is a list of all the functions included in this module :
 gsl_eigen_symm_alloc($n)  This function returns a workspace for computing
eigenvalues of nbyn real symmetric matrices.
 gsl_eigen_symm_free($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 nbyn 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 nbyn 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 nbyn 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 nbyn 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 1by1 real eigenvalues and
2by2 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 nbyn 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 nbyn 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 nbyn real generalized symmetricdefinite
eigensystems.
 gsl_eigen_gensymm_free($w)  This function frees the memory associated
with the workspace $w.
 gsl_eigen_gensymm($A, $B, $eval, $w)  This function computes the
eigenvalues of the real generalized symmetricdefinite matrix pair ($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 nbyn real generalized
symmetricdefinite eigensystems.
 gsl_eigen_gensymmv_free($w)  This function frees the memory associated
with the workspace $w.
 gsl_eigen_gensymmv($A, $B, $eval, $evec, $w)  This function computes the
eigenvalues and eigenvectors of the real generalized symmetricdefinite
matrix pair ($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 nbyn complex generalized hermitiandefinite
eigensystems.
 gsl_eigen_genherm_free($w)  This function frees the memory associated
with the workspace $w.
 gsl_eigen_genherm($A, $B, $eval, $w)  This function computes the
eigenvalues of the complex generalized hermitiandefinite matrix pair ($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 nbyn complex generalized
hermitiandefinite eigensystems.
 gsl_eigen_genhermv_free($w)  This function frees the memory associated
with the workspace $w.
 gsl_eigen_genhermv($A, $B, $eval, $evec, $w)  This function computes the
eigenvalues and eigenvectors of the complex generalized hermitiandefinite
matrix pair ($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 nbyn 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 1by1 and 2by2 blocks on its diagonal. 1by1 blocks correspond to
real eigenvalues, and 2by2 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 nonnegative elements
on its diagonal. Any 2by2 blocks in S will correspond to a 2by2 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
nonnegative. 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 nbyn 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, $w)  This function computes
eigenvalues and right eigenvectors of the nbyn real generalized
nonsymmetric matrix pair ($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 informations on the functions, we refer you to the GSL offcial
documentation: <http://www.gnu.org/software/gsl/manual/html_node/>
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";
Jonathan "Duke" Leto <jonathan@leto.net> and Thierry Moisan
<thierry.moisan@gmail.com>
Copyright (C) 20082011 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.
Visit the GSP FreeBSD Man Page Interface. Output converted with ManDoc. 