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<title>Real Generalized Symmetric-Definite Eigensystems - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">14.4 Real Generalized Symmetric-Definite Eigensystems</h3>
<p><a name="index-generalized-symmetric-eigensystems-1358"></a>
The real generalized symmetric-definite eigenvalue problem is to find
eigenvalues \lambda and eigenvectors x such that
<pre class="example"> A x = \lambda B x
</pre>
<p>where A and B are symmetric matrices, and B is
positive-definite. This problem reduces to the standard symmetric
eigenvalue problem by applying the Cholesky decomposition to B:
<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>
<p>Therefore, the problem becomes C y = \lambda y where
<!-- {$C = L^{-1} A L^{-t}$} -->
C = L^{-1} A L^{-t}
is symmetric, and y = L^t x. The standard
symmetric eigensolver can be applied to the matrix C.
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.
<div class="defun">
— Function: gsl_eigen_gensymm_workspace * <b>gsl_eigen_gensymm_alloc</b> (<var>const size_t n</var>)<var><a name="index-gsl_005feigen_005fgensymm_005falloc-1359"></a></var><br>
<blockquote><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 O(2n).
</p></blockquote></div>
<div class="defun">
— Function: void <b>gsl_eigen_gensymm_free</b> (<var>gsl_eigen_gensymm_workspace * w</var>)<var><a name="index-gsl_005feigen_005fgensymm_005ffree-1360"></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_gensymm</b> (<var>gsl_matrix * A, gsl_matrix * B, gsl_vector * eval, gsl_eigen_gensymm_workspace * w</var>)<var><a name="index-gsl_005feigen_005fgensymm-1361"></a></var><br>
<blockquote><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></blockquote></div>
<div class="defun">
— Function: gsl_eigen_gensymmv_workspace * <b>gsl_eigen_gensymmv_alloc</b> (<var>const size_t n</var>)<var><a name="index-gsl_005feigen_005fgensymmv_005falloc-1362"></a></var><br>
<blockquote><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 O(4n).
</p></blockquote></div>
<div class="defun">
— Function: void <b>gsl_eigen_gensymmv_free</b> (<var>gsl_eigen_gensymmv_workspace * w</var>)<var><a name="index-gsl_005feigen_005fgensymmv_005ffree-1363"></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_gensymmv</b> (<var>gsl_matrix * A, gsl_matrix * B, gsl_vector * eval, gsl_matrix * evec, gsl_eigen_gensymmv_workspace * w</var>)<var><a name="index-gsl_005feigen_005fgensymmv-1364"></a></var><br>
<blockquote><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></blockquote></div>
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