GNU Scientific Library Reference Manual - Third Edition (v1.12)by M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. Rossi Paperback (6"x9"), 592 pages, 60 figures ISBN 0954612078 RRP £24.95 ($39.95) |

## 14.6 Real Generalized Nonsymmetric Eigensystems

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).

__Function:__gsl_eigen_gen_workspace ***gsl_eigen_gen_alloc***(const size_t*`n`)- This function allocates a workspace for computing eigenvalues of
`n`-by-`n`real generalized nonsymmetric eigensystems. The size of the workspace is O(n).

__Function:__void**gsl_eigen_gen_free***(gsl_eigen_gen_workspace **`w`)- This function frees the memory associated with the workspace
`w`.

__Function:__void**gsl_eigen_gen_params***(const int*`compute_s`, const int`compute_t`, const int`balance`, gsl_eigen_gen_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.

__Function:__int**gsl_eigen_gen***(gsl_matrix **`A`, gsl_matrix *`B`, gsl_vector_complex *`alpha`, gsl_vector *`beta`, gsl_eigen_gen_workspace *`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. 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.

__Function:__int**gsl_eigen_gen_QZ***(gsl_matrix **`A`, gsl_matrix *`B`, gsl_vector_complex *`alpha`, gsl_vector *`beta`, gsl_matrix *`Q`, gsl_matrix *`Z`, gsl_eigen_gen_workspace *`w`)- 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.

__Function:__gsl_eigen_genv_workspace ***gsl_eigen_genv_alloc***(const size_t*`n`)- 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).

__Function:__void**gsl_eigen_genv_free***(gsl_eigen_genv_workspace **`w`)- This function frees the memory associated with the workspace
`w`.

__Function:__int**gsl_eigen_genv***(gsl_matrix **`A`, gsl_matrix *`B`, gsl_vector_complex *`alpha`, gsl_vector *`beta`, gsl_matrix_complex *`evec`, gsl_eigen_genv_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.

__Function:__int**gsl_eigen_genv_QZ***(gsl_matrix **`A`, gsl_matrix *`B`, gsl_vector_complex *`alpha`, gsl_vector *`beta`, gsl_matrix_complex *`evec`, gsl_matrix *`Q`, gsl_matrix *`Z`, gsl_eigen_genv_workspace *`w`)- 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.

ISBN 0954612078 | GNU Scientific Library Reference Manual - Third Edition (v1.12) | See the print edition |