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<a name="Real-Generalized-Symmetric_002dDefinite-Eigensystems"></a>
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<p>
Next: <a href="Complex-Generalized-Hermitian_002dDefinite-Eigensystems.html#Complex-Generalized-Hermitian_002dDefinite-Eigensystems" accesskey="n" rel="next">Complex Generalized Hermitian-Definite Eigensystems</a>, Previous: <a href="Real-Nonsymmetric-Matrices.html#Real-Nonsymmetric-Matrices" accesskey="p" rel="previous">Real Nonsymmetric Matrices</a>, Up: <a href="Eigensystems.html#Eigensystems" accesskey="u" rel="up">Eigensystems</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Real-Generalized-Symmetric_002dDefinite-Eigensystems-1"></a>
<h3 class="section">15.4 Real Generalized Symmetric-Definite Eigensystems</h3>
<a name="index-generalized-symmetric-eigensystems"></a>
<p>The real generalized symmetric-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 symmetric matrices, and <em>B</em> is
positive-definite. This problem reduces to the standard symmetric
eigenvalue problem by applying the Cholesky decomposition to <em>B</em>:
</p><div class="example">
<pre class="example"> A x = \lambda B x
A x = \lambda L L^t x
( L^{-1} A L^{-t} ) L^t x = \lambda L^t x
</pre></div>
<p>Therefore, the problem becomes <em>C y = \lambda y</em> where
<em>C = L^{-1} A L^{-t}</em>
is symmetric, and <em>y = L^t x</em>. The standard
symmetric 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 and eigenvectors
of the generalized symmetric-definite eigenproblem are always real.
</p>
<dl>
<dt><a name="index-gsl_005feigen_005fgensymm_005falloc"></a>Function: <em>gsl_eigen_gensymm_workspace *</em> <strong>gsl_eigen_gensymm_alloc</strong> <em>(const size_t <var>n</var>)</em></dt>
<dd><a name="index-gsl_005feigen_005fgensymm_005fworkspace"></a>
<p>This function allocates a workspace for computing eigenvalues of
<var>n</var>-by-<var>n</var> real generalized symmetric-definite eigensystems. The
size of the workspace is <em>O(2n)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005feigen_005fgensymm_005ffree"></a>Function: <em>void</em> <strong>gsl_eigen_gensymm_free</strong> <em>(gsl_eigen_gensymm_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_005fgensymm"></a>Function: <em>int</em> <strong>gsl_eigen_gensymm</strong> <em>(gsl_matrix * <var>A</var>, gsl_matrix * <var>B</var>, gsl_vector * <var>eval</var>, gsl_eigen_gensymm_workspace * <var>w</var>)</em></dt>
<dd><p>This function computes the eigenvalues of the real generalized
symmetric-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_005fgensymmv_005falloc"></a>Function: <em>gsl_eigen_gensymmv_workspace *</em> <strong>gsl_eigen_gensymmv_alloc</strong> <em>(const size_t <var>n</var>)</em></dt>
<dd><a name="index-gsl_005feigen_005fgensymmv_005fworkspace"></a>
<p>This function allocates a workspace for computing eigenvalues and
eigenvectors of <var>n</var>-by-<var>n</var> real generalized symmetric-definite
eigensystems. The size of the workspace is <em>O(4n)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005feigen_005fgensymmv_005ffree"></a>Function: <em>void</em> <strong>gsl_eigen_gensymmv_free</strong> <em>(gsl_eigen_gensymmv_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_005fgensymmv"></a>Function: <em>int</em> <strong>gsl_eigen_gensymmv</strong> <em>(gsl_matrix * <var>A</var>, gsl_matrix * <var>B</var>, gsl_vector * <var>eval</var>, gsl_matrix * <var>evec</var>, gsl_eigen_gensymmv_workspace * <var>w</var>)</em></dt>
<dd><p>This function computes the eigenvalues and eigenvectors of the real
generalized symmetric-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="Complex-Generalized-Hermitian_002dDefinite-Eigensystems.html#Complex-Generalized-Hermitian_002dDefinite-Eigensystems" accesskey="n" rel="next">Complex Generalized Hermitian-Definite Eigensystems</a>, Previous: <a href="Real-Nonsymmetric-Matrices.html#Real-Nonsymmetric-Matrices" accesskey="p" rel="previous">Real Nonsymmetric Matrices</a>, Up: <a href="Eigensystems.html#Eigensystems" accesskey="u" rel="up">Eigensystems</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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