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) |

## 14.3 Real Nonsymmetric Matrices

The solution of the real nonsymmetric eigensystem problem for a matrix A involves computing the Schur decomposition

A = Z T Z^T

where Z is an orthogonal matrix of Schur vectors and T, the Schur form, is quasi upper triangular with diagonal 1-by-1 blocks which are real eigenvalues of A, and diagonal 2-by-2 blocks whose eigenvalues are complex conjugate eigenvalues of A. The algorithm used is the double-shift Francis method.

__Function:__gsl_eigen_nonsymm_workspace ***gsl_eigen_nonsymm_alloc***(const size_t*`n`)- This function allocates a workspace for computing eigenvalues of
`n`-by-`n`real nonsymmetric matrices. The size of the workspace is O(2n).

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

__Function:__void**gsl_eigen_nonsymm_params***(const int*`compute_t`, const int`balance`, gsl_eigen_nonsymm_workspace *`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). Computing the full Schur form T requires approximately 1.5--2 times the number of flops.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. See 13.14 for more information. Note that the balancing transformation does not preserve the orthogonality of the Schur vectors, so if you wish to compute the Schur vectors with`gsl_eigen_nonsymm_Z`

you will obtain the Schur vectors of the balanced matrix instead of the original matrix. The relationship will beT = Q^t D^(-1) A D Q

where

`Q`is the matrix of Schur vectors for the balanced matrix, and`D`is the balancing transformation. Then`gsl_eigen_nonsymm_Z`

will compute a matrix`Z`which satisfiesT = Z^(-1) A Z

with Z = D Q. Note that

`Z`will not be orthogonal. For this reason, balancing is not performed by default.

__Function:__int**gsl_eigen_nonsymm***(gsl_matrix **`A`, gsl_vector_complex *`eval`, gsl_eigen_nonsymm_workspace *`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 and the number of converged eigenvalues is stored in`w->n_evals`

. The converged eigenvalues are stored in the beginning of`eval`.

__Function:__int**gsl_eigen_nonsymm_Z***(gsl_matrix **`A`, gsl_vector_complex *`eval`, gsl_matrix *`Z`, gsl_eigen_nonsymm_workspace *`w`)- This function is identical to
`gsl_eigen_nonsymm`

except that it also computes the Schur vectors and stores them into`Z`.

__Function:__gsl_eigen_nonsymmv_workspace ***gsl_eigen_nonsymmv_alloc***(const size_t*`n`)- This function allocates a workspace for computing eigenvalues and
eigenvectors of
`n`-by-`n`real nonsymmetric matrices. The size of the workspace is O(5n).

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

__Function:__int**gsl_eigen_nonsymmv***(gsl_matrix **`A`, gsl_vector_complex *`eval`, gsl_matrix_complex *`evec`, gsl_eigen_nonsymmv_workspace *`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.

__Function:__int**gsl_eigen_nonsymmv_Z***(gsl_matrix **`A`, gsl_vector_complex *`eval`, gsl_matrix_complex *`evec`, gsl_matrix *`Z`, gsl_eigen_nonsymmv_workspace *`w`)- This function is identical to
`gsl_eigen_nonsymmv`

except that it also saves the Schur vectors into`Z`.

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