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<title>Real Symmetric Matrices - GNU Scientific Library -- Reference Manual</title>
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Next: <a rel="next" accesskey="n" href="Complex-Hermitian-Matrices.html">Complex Hermitian Matrices</a>,
Up: <a rel="up" accesskey="u" href="Eigensystems.html">Eigensystems</a>
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<h3 class="section">14.1 Real Symmetric Matrices</h3>
<p><a name="index-symmetric-matrix_002c-real_002c-eigensystem-1331"></a><a name="index-real-symmetric-matrix_002c-eigensystem-1332"></a>
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 \epsilon ||A||_2, where \epsilon is
the machine precision.
<div class="defun">
— Function: gsl_eigen_symm_workspace * <b>gsl_eigen_symm_alloc</b> (<var>const size_t n</var>)<var><a name="index-gsl_005feigen_005fsymm_005falloc-1333"></a></var><br>
<blockquote><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 O(2n).
</p></blockquote></div>
<div class="defun">
— Function: void <b>gsl_eigen_symm_free</b> (<var>gsl_eigen_symm_workspace * w</var>)<var><a name="index-gsl_005feigen_005fsymm_005ffree-1334"></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_symm</b> (<var>gsl_matrix * A, gsl_vector * eval, gsl_eigen_symm_workspace * w</var>)<var><a name="index-gsl_005feigen_005fsymm-1335"></a></var><br>
<blockquote><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></blockquote></div>
<div class="defun">
— Function: gsl_eigen_symmv_workspace * <b>gsl_eigen_symmv_alloc</b> (<var>const size_t n</var>)<var><a name="index-gsl_005feigen_005fsymmv_005falloc-1336"></a></var><br>
<blockquote><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 O(4n).
</p></blockquote></div>
<div class="defun">
— Function: void <b>gsl_eigen_symmv_free</b> (<var>gsl_eigen_symmv_workspace * w</var>)<var><a name="index-gsl_005feigen_005fsymmv_005ffree-1337"></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_symmv</b> (<var>gsl_matrix * A, gsl_vector * eval, gsl_matrix * evec, gsl_eigen_symmv_workspace * w</var>)<var><a name="index-gsl_005feigen_005fsymmv-1338"></a></var><br>
<blockquote><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></blockquote></div>
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