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<title>Complex Generalized Hermitian-Definite Eigensystems - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">15.5 Complex Generalized Hermitian-Definite Eigensystems</h3>
<p><a name="index-generalized-hermitian-definite-eigensystems-1440"></a>
The complex generalized hermitian-definite eigenvalue problem is to find
eigenvalues \lambda and eigenvectors x such that
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
<!-- {$C = L^{-1} A L^{-\dagger}$} -->
C = L^{-1} A L^{-H}
is hermitian, and
<!-- {$y = L^{\dagger} x$} -->
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.
<div class="defun">
— Function: gsl_eigen_genherm_workspace * <b>gsl_eigen_genherm_alloc</b> (<var>const size_t n</var>)<var><a name="index-gsl_005feigen_005fgenherm_005falloc-1441"></a></var><br>
<blockquote><p><a name="index-gsl_005feigen_005fgenherm_005fworkspace-1442"></a>This function allocates a workspace for computing eigenvalues of
<var>n</var>-by-<var>n</var> complex generalized hermitian-definite eigensystems. The
size of the workspace is O(3n).
</p></blockquote></div>
<div class="defun">
— Function: void <b>gsl_eigen_genherm_free</b> (<var>gsl_eigen_genherm_workspace * w</var>)<var><a name="index-gsl_005feigen_005fgenherm_005ffree-1443"></a></var><br>
<blockquote><p>This function frees the memory associated with the workspace <var>w</var>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_eigen_genherm</b> (<var>gsl_matrix_complex * A, gsl_matrix_complex * B, gsl_vector * eval, gsl_eigen_genherm_workspace * w</var>)<var><a name="index-gsl_005feigen_005fgenherm-1444"></a></var><br>
<blockquote><p>This function computes the eigenvalues of the complex generalized
hermitian-definite matrix pair (<var>A</var>, <var>B</var>), and stores them
in <var>eval</var>, using the method outlined above. On output, <var>B</var>
contains its Cholesky decomposition and <var>A</var> is destroyed.
</p></blockquote></div>
<div class="defun">
— Function: gsl_eigen_genhermv_workspace * <b>gsl_eigen_genhermv_alloc</b> (<var>const size_t n</var>)<var><a name="index-gsl_005feigen_005fgenhermv_005falloc-1445"></a></var><br>
<blockquote><p><a name="index-gsl_005feigen_005fgenhermv_005fworkspace-1446"></a>This function allocates a workspace for computing eigenvalues and
eigenvectors of <var>n</var>-by-<var>n</var> complex generalized hermitian-definite
eigensystems. The size of the workspace is O(5n).
</p></blockquote></div>
<div class="defun">
— Function: void <b>gsl_eigen_genhermv_free</b> (<var>gsl_eigen_genhermv_workspace * w</var>)<var><a name="index-gsl_005feigen_005fgenhermv_005ffree-1447"></a></var><br>
<blockquote><p>This function frees the memory associated with the workspace <var>w</var>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_eigen_genhermv</b> (<var>gsl_matrix_complex * A, gsl_matrix_complex * B, gsl_vector * eval, gsl_matrix_complex * evec, gsl_eigen_genhermv_workspace * w</var>)<var><a name="index-gsl_005feigen_005fgenhermv-1448"></a></var><br>
<blockquote><p>This function computes the eigenvalues and eigenvectors of the complex
generalized hermitian-definite matrix pair (<var>A</var>, <var>B</var>), and
stores them in <var>eval</var> and <var>evec</var> respectively. The computed
eigenvectors are normalized to have unit magnitude. On output,
<var>B</var> contains its Cholesky decomposition and <var>A</var> is destroyed.
</p></blockquote></div>
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