Next: Complex Hermitian Matrices, Up: Eigensystems [Index]

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.

Next: Complex Hermitian Matrices, Up: Eigensystems [Index]