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<a class="el" href="namespace_quant_lib.html">QuantLib</a>::<a class="el" href="class_quant_lib_1_1_symmetric_schur_decomposition.html">SymmetricSchurDecomposition</a></div>
<h1>SymmetricSchurDecomposition Class Reference</h1><!-- doxytag: class="QuantLib::SymmetricSchurDecomposition" --><code>#include <ql/math/matrixutilities/symmetricschurdecomposition.hpp></code>
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<a href="class_quant_lib_1_1_symmetric_schur_decomposition-members.html">List of all members.</a><hr><a name="_details"></a><h2>Detailed Description</h2>
symmetric threshold Jacobi algorithm.
<p>
Given a real symmetric matrix S, the Schur decomposition finds the eigenvalues and eigenvectors of S. If D is the diagonal matrix formed by the eigenvalues and U the unitarian matrix of the eigenvectors we can write the Schur decomposition as <p class="formulaDsp">
<img class="formulaDsp" alt="\[ S = U \cdot D \cdot U^T \, ,\]" src="form_145.png">
<p>
where <img class="formulaInl" alt="$ \cdot $" src="form_146.png"> is the standard matrix product and <img class="formulaInl" alt="$ ^T $" src="form_147.png"> is the transpose operator. This class implements the Schur decomposition using the symmetric threshold Jacobi algorithm. For details on the different Jacobi transfomations see "Matrix computation," second edition, by Golub and Van Loan, The Johns Hopkins University Press<p>
<dl compact><dt><b><a class="el" href="test.html#_test000039">Tests:</a></b></dt><dd>the correctness of the returned values is tested by checking their properties. </dd></dl>
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<tr><td class="memItemLeft" nowrap align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_symmetric_schur_decomposition.html#43ff4c0e9abb6877c224127f7b70e8e6">SymmetricSchurDecomposition</a> (const <a class="el" href="class_quant_lib_1_1_matrix.html">Matrix</a> &s)</td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="eb9833fe31a4739b5075f2a4cbd5dddd"></a><!-- doxytag: member="QuantLib::SymmetricSchurDecomposition::eigenvalues" ref="eb9833fe31a4739b5075f2a4cbd5dddd" args="() const " -->
const <a class="el" href="class_quant_lib_1_1_array.html">Array</a> & </td><td class="memItemRight" valign="bottom"><b>eigenvalues</b> () const </td></tr>
<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="41aa0134fde9e36a14c3a0938903e4b1"></a><!-- doxytag: member="QuantLib::SymmetricSchurDecomposition::eigenvectors" ref="41aa0134fde9e36a14c3a0938903e4b1" args="() const " -->
const <a class="el" href="class_quant_lib_1_1_matrix.html">Matrix</a> & </td><td class="memItemRight" valign="bottom"><b>eigenvectors</b> () const </td></tr>
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<hr><h2>Constructor & Destructor Documentation</h2>
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<td class="paramtype">const <a class="el" href="class_quant_lib_1_1_matrix.html">Matrix</a> & </td>
<td class="paramname"> <em>s</em> </td>
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<dl class="pre" compact><dt><b>Precondition:</b></dt><dd>s must be symmetric </dd></dl>
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