Next: Real Generalized Nonsymmetric Eigensystems, Previous: Real Generalized Symmetric-Definite Eigensystems, Up: Eigensystems [Index]

The complex generalized hermitian-definite eigenvalue problem is to find
eigenvalues *\lambda* and eigenvectors *x* such that

A x = \lambda B x

where *A* and *B* are hermitian matrices, and *B* is
positive-definite. Similarly to the real case, this can be reduced
to *C y = \lambda y* where
*C = L^{-1} A L^{-H}*
is hermitian, and
*y = L^H x*. The standard
hermitian eigensolver can be applied to the matrix *C*.
The resulting eigenvectors are backtransformed to find the
vectors of the original problem. The eigenvalues
of the generalized hermitian-definite eigenproblem are always real.

- Function:
*gsl_eigen_genherm_workspace ****gsl_eigen_genherm_alloc***(const size_t*`n`) -
This function allocates a workspace for computing eigenvalues of

`n`-by-`n`complex generalized hermitian-definite eigensystems. The size of the workspace is*O(3n)*.

- Function:
*void***gsl_eigen_genherm_free***(gsl_eigen_genherm_workspace **`w`) This function frees the memory associated with the workspace

`w`.

- Function:
*int***gsl_eigen_genherm***(gsl_matrix_complex **`A`, gsl_matrix_complex *`B`, gsl_vector *`eval`, gsl_eigen_genherm_workspace *`w`) This function computes the eigenvalues of the complex generalized hermitian-definite matrix pair (

`A`,`B`), and stores them in`eval`, using the method outlined above. On output,`B`contains its Cholesky decomposition and`A`is destroyed.

- Function:
*gsl_eigen_genhermv_workspace ****gsl_eigen_genhermv_alloc***(const size_t*`n`) -
This function allocates a workspace for computing eigenvalues and eigenvectors of

`n`-by-`n`complex generalized hermitian-definite eigensystems. The size of the workspace is*O(5n)*.

- Function:
*void***gsl_eigen_genhermv_free***(gsl_eigen_genhermv_workspace **`w`) This function frees the memory associated with the workspace

`w`.

- Function:
*int***gsl_eigen_genhermv***(gsl_matrix_complex **`A`, gsl_matrix_complex *`B`, gsl_vector *`eval`, gsl_matrix_complex *`evec`, gsl_eigen_genhermv_workspace *`w`) This function computes the eigenvalues and eigenvectors of the complex generalized hermitian-definite matrix pair (

`A`,`B`), and stores them in`eval`and`evec`respectively. The computed eigenvectors are normalized to have unit magnitude. On output,`B`contains its Cholesky decomposition and`A`is destroyed.