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) |

## 14.1 Real Symmetric Matrices

For real symmetric matrices, the library uses the symmetric bidiagonalization and QR reduction method. This is described in Golub & van Loan, section 8.3. The computed eigenvalues are accurate to an absolute accuracy of \epsilon ||A||_2, where \epsilon is the machine precision.

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

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

__Function:__int**gsl_eigen_symm***(gsl_matrix **`A`, gsl_vector *`eval`, gsl_eigen_symm_workspace *`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.

__Function:__gsl_eigen_symmv_workspace ***gsl_eigen_symmv_alloc***(const size_t*`n`)- This function allocates a workspace for computing eigenvalues and
eigenvectors of
`n`-by-`n`real symmetric matrices. The size of the workspace is O(4n).

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

__Function:__int**gsl_eigen_symmv***(gsl_matrix **`A`, gsl_vector *`eval`, gsl_matrix *`evec`, gsl_eigen_symmv_workspace *`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`. For example, the eigenvector in the first column corresponds to the first eigenvalue. The eigenvectors are guaranteed to be mutually orthogonal and normalised to unit magnitude.

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