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### 15.2 Complex Hermitian Matrices

For hermitian matrices, the library uses the complex form of the symmetric bidiagonalization and QR reduction method.

Function: gsl_eigen_herm_workspace * gsl_eigen_herm_alloc (const size_t n)

This function allocates a workspace for computing eigenvalues of n-by-n complex hermitian matrices. The size of the workspace is O(3n).

Function: void gsl_eigen_herm_free (gsl_eigen_herm_workspace * w)

This function frees the memory associated with the workspace w.

Function: int gsl_eigen_herm (gsl_matrix_complex * A, gsl_vector * eval, gsl_eigen_herm_workspace * w)

This function computes the eigenvalues of the complex hermitian 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 imaginary parts of the diagonal are assumed to be zero and are not referenced. The eigenvalues are stored in the vector eval and are unordered.

Function: gsl_eigen_hermv_workspace * gsl_eigen_hermv_alloc (const size_t n)

This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n complex hermitian matrices. The size of the workspace is O(5n).

Function: void gsl_eigen_hermv_free (gsl_eigen_hermv_workspace * w)

This function frees the memory associated with the workspace w.

Function: int gsl_eigen_hermv (gsl_matrix_complex * A, gsl_vector * eval, gsl_matrix_complex * evec, gsl_eigen_hermv_workspace * w)

This function computes the eigenvalues and eigenvectors of the complex hermitian 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 imaginary parts of the diagonal are assumed to be zero and are not referenced. The eigenvalues are stored in the vector eval and are unordered. The corresponding complex 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.

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