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<title>GNU Scientific Library – Reference Manual: Real Symmetric Matrices</title>
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<a name="Real-Symmetric-Matrices"></a>
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Next: <a href="Complex-Hermitian-Matrices.html#Complex-Hermitian-Matrices" accesskey="n" rel="next">Complex Hermitian 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-Symmetric-Matrices-1"></a>
<h3 class="section">15.1 Real Symmetric Matrices</h3>
<a name="index-symmetric-matrix_002c-real_002c-eigensystem"></a>
<a name="index-real-symmetric-matrix_002c-eigensystem"></a>
<p>For real symmetric matrices, the library uses the symmetric
bidiagonalization and QR reduction method. This is described in Golub
& van Loan, section 8.3. The computed eigenvalues are accurate to an
absolute accuracy of <em>\epsilon ||A||_2</em>, where <em>\epsilon</em> is
the machine precision.
</p>
<dl>
<dt><a name="index-gsl_005feigen_005fsymm_005falloc"></a>Function: <em>gsl_eigen_symm_workspace *</em> <strong>gsl_eigen_symm_alloc</strong> <em>(const size_t <var>n</var>)</em></dt>
<dd><a name="index-gsl_005feigen_005fsymm_005fworkspace"></a>
<p>This function allocates a workspace for computing eigenvalues of
<var>n</var>-by-<var>n</var> real symmetric matrices. The size of the workspace
is <em>O(2n)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005feigen_005fsymm_005ffree"></a>Function: <em>void</em> <strong>gsl_eigen_symm_free</strong> <em>(gsl_eigen_symm_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_005fsymm"></a>Function: <em>int</em> <strong>gsl_eigen_symm</strong> <em>(gsl_matrix * <var>A</var>, gsl_vector * <var>eval</var>, gsl_eigen_symm_workspace * <var>w</var>)</em></dt>
<dd><p>This function computes the eigenvalues of the real symmetric matrix
<var>A</var>. Additional workspace of the appropriate size must be provided
in <var>w</var>. The diagonal and lower triangular part of <var>A</var> are
destroyed during the computation, but the strict upper triangular part
is not referenced. The eigenvalues are stored in the vector <var>eval</var>
and are unordered.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005feigen_005fsymmv_005falloc"></a>Function: <em>gsl_eigen_symmv_workspace *</em> <strong>gsl_eigen_symmv_alloc</strong> <em>(const size_t <var>n</var>)</em></dt>
<dd><a name="index-gsl_005feigen_005fsymmv_005fworkspace"></a>
<p>This function allocates a workspace for computing eigenvalues and
eigenvectors of <var>n</var>-by-<var>n</var> real symmetric matrices. The size of
the workspace is <em>O(4n)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005feigen_005fsymmv_005ffree"></a>Function: <em>void</em> <strong>gsl_eigen_symmv_free</strong> <em>(gsl_eigen_symmv_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_005fsymmv"></a>Function: <em>int</em> <strong>gsl_eigen_symmv</strong> <em>(gsl_matrix * <var>A</var>, gsl_vector * <var>eval</var>, gsl_matrix * <var>evec</var>, gsl_eigen_symmv_workspace * <var>w</var>)</em></dt>
<dd><p>This function computes the eigenvalues and eigenvectors of the real
symmetric matrix <var>A</var>. Additional workspace of the appropriate size
must be provided in <var>w</var>. The diagonal and lower triangular part of
<var>A</var> are destroyed during the computation, but the strict upper
triangular part is not referenced. The eigenvalues are stored in the
vector <var>eval</var> and are unordered. The corresponding eigenvectors are
stored in the columns of the matrix <var>evec</var>. 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.
</p></dd></dl>
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