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<a name="Complex-Generalized-Hermitian_002dDefinite-Eigensystems"></a>
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Next: <a href="Real-Generalized-Nonsymmetric-Eigensystems.html#Real-Generalized-Nonsymmetric-Eigensystems" accesskey="n" rel="next">Real Generalized Nonsymmetric Eigensystems</a>, Previous: <a href="Real-Generalized-Symmetric_002dDefinite-Eigensystems.html#Real-Generalized-Symmetric_002dDefinite-Eigensystems" accesskey="p" rel="previous">Real Generalized Symmetric-Definite Eigensystems</a>, Up: <a href="Eigensystems.html#Eigensystems" accesskey="u" rel="up">Eigensystems</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Complex-Generalized-Hermitian_002dDefinite-Eigensystems-1"></a>
<h3 class="section">15.5 Complex Generalized Hermitian-Definite Eigensystems</h3>
<a name="index-generalized-hermitian-definite-eigensystems"></a>

<p>The complex generalized hermitian-definite eigenvalue problem is to find
eigenvalues <em>\lambda</em> and eigenvectors <em>x</em> such that
</p><div class="example">
<pre class="example">A x = \lambda B x
</pre></div>
<p>where <em>A</em> and <em>B</em> are hermitian matrices, and <em>B</em> is
positive-definite. Similarly to the real case, this can be reduced
to <em>C y = \lambda y</em> where
<em>C = L^{-1} A L^{-H}</em>
is hermitian, and
<em>y = L^H x</em>. The standard
hermitian eigensolver can be applied to the matrix <em>C</em>.
The resulting eigenvectors are backtransformed to find the
vectors of the original problem. The eigenvalues
of the generalized hermitian-definite eigenproblem are always real.
</p>
<dl>
<dt><a name="index-gsl_005feigen_005fgenherm_005falloc"></a>Function: <em>gsl_eigen_genherm_workspace *</em> <strong>gsl_eigen_genherm_alloc</strong> <em>(const size_t <var>n</var>)</em></dt>
<dd><a name="index-gsl_005feigen_005fgenherm_005fworkspace"></a>
<p>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 <em>O(3n)</em>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005feigen_005fgenherm_005ffree"></a>Function: <em>void</em> <strong>gsl_eigen_genherm_free</strong> <em>(gsl_eigen_genherm_workspace * <var>w</var>)</em></dt>
<dd><p>This function frees the memory associated with the workspace <var>w</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005feigen_005fgenherm"></a>Function: <em>int</em> <strong>gsl_eigen_genherm</strong> <em>(gsl_matrix_complex * <var>A</var>, gsl_matrix_complex * <var>B</var>, gsl_vector * <var>eval</var>, gsl_eigen_genherm_workspace * <var>w</var>)</em></dt>
<dd><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></dd></dl>

<dl>
<dt><a name="index-gsl_005feigen_005fgenhermv_005falloc"></a>Function: <em>gsl_eigen_genhermv_workspace *</em> <strong>gsl_eigen_genhermv_alloc</strong> <em>(const size_t <var>n</var>)</em></dt>
<dd><a name="index-gsl_005feigen_005fgenhermv_005fworkspace"></a>
<p>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 <em>O(5n)</em>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005feigen_005fgenhermv_005ffree"></a>Function: <em>void</em> <strong>gsl_eigen_genhermv_free</strong> <em>(gsl_eigen_genhermv_workspace * <var>w</var>)</em></dt>
<dd><p>This function frees the memory associated with the workspace <var>w</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005feigen_005fgenhermv"></a>Function: <em>int</em> <strong>gsl_eigen_genhermv</strong> <em>(gsl_matrix_complex * <var>A</var>, gsl_matrix_complex * <var>B</var>, gsl_vector * <var>eval</var>, gsl_matrix_complex * <var>evec</var>, gsl_eigen_genhermv_workspace * <var>w</var>)</em></dt>
<dd><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></dd></dl>

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Next: <a href="Real-Generalized-Nonsymmetric-Eigensystems.html#Real-Generalized-Nonsymmetric-Eigensystems" accesskey="n" rel="next">Real Generalized Nonsymmetric Eigensystems</a>, Previous: <a href="Real-Generalized-Symmetric_002dDefinite-Eigensystems.html#Real-Generalized-Symmetric_002dDefinite-Eigensystems" accesskey="p" rel="previous">Real Generalized Symmetric-Definite Eigensystems</a>, Up: <a href="Eigensystems.html#Eigensystems" accesskey="u" rel="up">Eigensystems</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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