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<div class="title">RealQZ< _MatrixType > Class Template Reference<div class="ingroups"><a class="el" href="group__Eigenvalues__Module.html">Eigenvalues module</a></div></div> </div>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><h3>template<typename _MatrixType><br/>
class Eigen::RealQZ< _MatrixType ></h3>
<p>Performs a real QZ decomposition of a pair of square matrices. </p>
<p>This is defined in the Eigenvalues module.</p>
<div class="fragment"><div class="line"><span class="preprocessor">#include <Eigen/Eigenvalues></span> </div>
</div><!-- fragment --><dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">_MatrixType</td><td>the type of the matrix of which we are computing the real QZ decomposition; this is expected to be an instantiation of the <a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> class template.</td></tr>
</table>
</dd>
</dl>
<p>Given a real square matrices A and B, this class computes the real QZ decomposition: <img class="formulaInl" alt="$ A = Q S Z $" src="form_89.png"/>, <img class="formulaInl" alt="$ B = Q T Z $" src="form_90.png"/> where Q and Z are real orthogonal matrixes, T is upper-triangular matrix, and S is upper quasi-triangular matrix. An orthogonal matrix is a matrix whose inverse is equal to its transpose, <img class="formulaInl" alt="$ U^{-1} = U^T $" src="form_91.png"/>. A quasi-triangular matrix is a block-triangular matrix whose diagonal consists of 1-by-1 blocks and 2-by-2 blocks where further reduction is impossible due to complex eigenvalues.</p>
<p>The eigenvalues of the pencil <img class="formulaInl" alt="$ A - z B $" src="form_92.png"/> can be obtained from 1x1 and 2x2 blocks on the diagonals of S and T.</p>
<p>Call the function <a class="el" href="classEigen_1_1RealQZ.html#a75cc509be2fa23cde7371b79ebd9f770" title="Computes QZ decomposition of given matrix. ">compute()</a> to compute the real QZ decomposition of a given pair of matrices. Alternatively, you can use the <a class="el" href="classEigen_1_1RealQZ.html#ab81ec305afdcbf94ed288d382710e8d7" title="Constructor; computes real QZ decomposition of given matrices. ">RealQZ(const MatrixType& B, const MatrixType& B, bool computeQZ)</a> constructor which computes the real QZ decomposition at construction time. Once the decomposition is computed, you can use the <a class="el" href="classEigen_1_1RealQZ.html#af0356bea58e012ff54177cea8a340f64" title="Returns matrix S in the QZ decomposition. ">matrixS()</a>, <a class="el" href="classEigen_1_1RealQZ.html#a0d31900234ef9fea5751ce8ea693d71f" title="Returns matrix S in the QZ decomposition. ">matrixT()</a>, <a class="el" href="classEigen_1_1RealQZ.html#a498541357c143f345f0af5d6a6b9b3c3" title="Returns matrix Q in the QZ decomposition. ">matrixQ()</a> and <a class="el" href="classEigen_1_1RealQZ.html#a0f5765f1177b790b663281635e15e73f" title="Returns matrix Z in the QZ decomposition. ">matrixZ()</a> functions to retrieve the matrices S, T, Q and Z in the decomposition. If computeQZ==false, some time is saved by not computing matrices Q and Z.</p>
<p>Example: </p>
<div class="fragment"><div class="line"><a class="code" href="group__matrixtypedefs.html#gabab09c32e96cfa9829a88400627af162">MatrixXf</a> A = <a class="code" href="classEigen_1_1DenseBase.html#a8e759dafdd9ecc446d397b7f5435f60a">MatrixXf::Random</a>(4,4);</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#gabab09c32e96cfa9829a88400627af162">MatrixXf</a> B = <a class="code" href="classEigen_1_1DenseBase.html#a8e759dafdd9ecc446d397b7f5435f60a">MatrixXf::Random</a>(4,4);</div>
<div class="line">RealQZ<MatrixXf> qz(4); <span class="comment">// preallocate space for 4x4 matrices</span></div>
<div class="line">qz.compute(A,B); <span class="comment">// A = Q S Z, B = Q T Z</span></div>
<div class="line"></div>
<div class="line"><span class="comment">// print original matrices and result of decomposition</span></div>
<div class="line">cout << <span class="stringliteral">"A:\n"</span> << A << <span class="stringliteral">"\n"</span> << <span class="stringliteral">"B:\n"</span> << B << <span class="stringliteral">"\n"</span>;</div>
<div class="line">cout << <span class="stringliteral">"S:\n"</span> << qz.matrixS() << <span class="stringliteral">"\n"</span> << <span class="stringliteral">"T:\n"</span> << qz.matrixT() << <span class="stringliteral">"\n"</span>;</div>
<div class="line">cout << <span class="stringliteral">"Q:\n"</span> << qz.matrixQ() << <span class="stringliteral">"\n"</span> << <span class="stringliteral">"Z:\n"</span> << qz.matrixZ() << <span class="stringliteral">"\n"</span>;</div>
<div class="line"></div>
<div class="line"><span class="comment">// verify precision</span></div>
<div class="line">cout << <span class="stringliteral">"\nErrors:"</span></div>
<div class="line"> << <span class="stringliteral">"\n|A-QSZ|: "</span> << (A-qz.matrixQ()*qz.matrixS()*qz.matrixZ()).norm()</div>
<div class="line"> << <span class="stringliteral">", |B-QTZ|: "</span> << (B-qz.matrixQ()*qz.matrixT()*qz.matrixZ()).norm()</div>
<div class="line"> << <span class="stringliteral">"\n|QQ* - I|: "</span> << (qz.matrixQ()*qz.matrixQ().adjoint() - <a class="code" href="classEigen_1_1MatrixBase.html#a0650b65c6ae6c3d19a138b72a6d68568">MatrixXf::Identity</a>(4,4)).norm()</div>
<div class="line"> << <span class="stringliteral">", |ZZ* - I|: "</span> << (qz.matrixZ()*qz.matrixZ().adjoint() - <a class="code" href="classEigen_1_1MatrixBase.html#a0650b65c6ae6c3d19a138b72a6d68568">MatrixXf::Identity</a>(4,4)).norm()</div>
<div class="line"> << <span class="stringliteral">"\n"</span>;</div>
</div><!-- fragment --><p> Output: </p>
<div class="fragment"><div class="line">A:</div>
<div class="line"> 0.68 0.823 -0.444 -0.27</div>
<div class="line"> -0.211 -0.605 0.108 0.0268</div>
<div class="line"> 0.566 -0.33 -0.0452 0.904</div>
<div class="line"> 0.597 0.536 0.258 0.832</div>
<div class="line">B:</div>
<div class="line"> 0.271 -0.967 -0.687 0.998</div>
<div class="line"> 0.435 -0.514 -0.198 -0.563</div>
<div class="line">-0.717 -0.726 -0.74 0.0259</div>
<div class="line"> 0.214 0.608 -0.782 0.678</div>
<div class="line">S:</div>
<div class="line"> 0.927 -0.928 0.643 -0.227</div>
<div class="line">-0.594 0.36 0.146 -0.606</div>
<div class="line"> 0 0 -0.398 -0.164</div>
<div class="line"> 0 0 0 -1.12</div>
<div class="line">T:</div>
<div class="line"> 1.51 0.278 -0.238 0.501</div>
<div class="line"> 0 -1.04 0.519 -0.239</div>
<div class="line"> 0 0 -1.25 0.438</div>
<div class="line"> 0 0 0 0.746</div>
<div class="line">Q:</div>
<div class="line"> 0.603 0.011 0.552 0.576</div>
<div class="line">-0.142 0.243 0.761 -0.585</div>
<div class="line"> 0.092 -0.958 0.152 -0.223</div>
<div class="line"> 0.78 0.149 -0.306 -0.526</div>
<div class="line">Z:</div>
<div class="line"> 0.284 0.26 -0.696 0.606</div>
<div class="line">-0.918 -0.108 -0.38 0.0406</div>
<div class="line">-0.269 0.783 0.462 0.32</div>
<div class="line">-0.0674 -0.555 0.398 0.727</div>
<div class="line"></div>
<div class="line">Errors:</div>
<div class="line">|A-QSZ|: 1.13e-06, |B-QTZ|: 1.81e-06</div>
<div class="line">|QQ* - I|: 1.01e-06, |ZZ* - I|: 7.02e-07</div>
</div><!-- fragment --><dl class="section note"><dt>Note</dt><dd>The implementation is based on the algorithm in "Matrix Computations" by Gene H. Golub and Charles F. Van Loan, and a paper "An algorithm for
generalized eigenvalue problems" by C.B.Moler and G.W.Stewart.</dd></dl>
<dl class="section see"><dt>See Also</dt><dd>class <a class="el" href="classEigen_1_1RealSchur.html" title="Performs a real Schur decomposition of a square matrix. ">RealSchur</a>, class <a class="el" href="classEigen_1_1ComplexSchur.html" title="Performs a complex Schur decomposition of a real or complex square matrix. ">ComplexSchur</a>, class <a class="el" href="classEigen_1_1EigenSolver.html" title="Computes eigenvalues and eigenvectors of general matrices. ">EigenSolver</a>, class <a class="el" href="classEigen_1_1ComplexEigenSolver.html" title="Computes eigenvalues and eigenvectors of general complex matrices. ">ComplexEigenSolver</a> </dd></dl>
</div><table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="pub-methods"></a>
Public Member Functions</h2></td></tr>
<tr class="memitem:a75cc509be2fa23cde7371b79ebd9f770"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1RealQZ.html">RealQZ</a> & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1RealQZ.html#a75cc509be2fa23cde7371b79ebd9f770">compute</a> (const MatrixType &A, const MatrixType &B, bool computeQZ=true)</td></tr>
<tr class="memdesc:a75cc509be2fa23cde7371b79ebd9f770"><td class="mdescLeft"> </td><td class="mdescRight">Computes QZ decomposition of given matrix. <a href="#a75cc509be2fa23cde7371b79ebd9f770">More...</a><br/></td></tr>
<tr class="separator:a75cc509be2fa23cde7371b79ebd9f770"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a0c06d5c2034ebb329c54235369643ad2"><td class="memItemLeft" align="right" valign="top"><a class="el" href="group__enums.html#ga51bc1ac16f26ebe51eae1abb77bd037b">ComputationInfo</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1RealQZ.html#a0c06d5c2034ebb329c54235369643ad2">info</a> () const </td></tr>
<tr class="memdesc:a0c06d5c2034ebb329c54235369643ad2"><td class="mdescLeft"> </td><td class="mdescRight">Reports whether previous computation was successful. <a href="#a0c06d5c2034ebb329c54235369643ad2">More...</a><br/></td></tr>
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<tr class="memitem:ae231768dac8df88971dc1871f6493571"><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="ae231768dac8df88971dc1871f6493571"></a>
Index </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1RealQZ.html#ae231768dac8df88971dc1871f6493571">iterations</a> () const </td></tr>
<tr class="memdesc:ae231768dac8df88971dc1871f6493571"><td class="mdescLeft"> </td><td class="mdescRight">Returns number of performed QR-like iterations. <br/></td></tr>
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<tr class="memitem:a498541357c143f345f0af5d6a6b9b3c3"><td class="memItemLeft" align="right" valign="top">const MatrixType & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1RealQZ.html#a498541357c143f345f0af5d6a6b9b3c3">matrixQ</a> () const </td></tr>
<tr class="memdesc:a498541357c143f345f0af5d6a6b9b3c3"><td class="mdescLeft"> </td><td class="mdescRight">Returns matrix Q in the QZ decomposition. <a href="#a498541357c143f345f0af5d6a6b9b3c3">More...</a><br/></td></tr>
<tr class="separator:a498541357c143f345f0af5d6a6b9b3c3"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:af0356bea58e012ff54177cea8a340f64"><td class="memItemLeft" align="right" valign="top">const MatrixType & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1RealQZ.html#af0356bea58e012ff54177cea8a340f64">matrixS</a> () const </td></tr>
<tr class="memdesc:af0356bea58e012ff54177cea8a340f64"><td class="mdescLeft"> </td><td class="mdescRight">Returns matrix S in the QZ decomposition. <a href="#af0356bea58e012ff54177cea8a340f64">More...</a><br/></td></tr>
<tr class="separator:af0356bea58e012ff54177cea8a340f64"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a0d31900234ef9fea5751ce8ea693d71f"><td class="memItemLeft" align="right" valign="top">const MatrixType & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1RealQZ.html#a0d31900234ef9fea5751ce8ea693d71f">matrixT</a> () const </td></tr>
<tr class="memdesc:a0d31900234ef9fea5751ce8ea693d71f"><td class="mdescLeft"> </td><td class="mdescRight">Returns matrix S in the QZ decomposition. <a href="#a0d31900234ef9fea5751ce8ea693d71f">More...</a><br/></td></tr>
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<tr class="memitem:a0f5765f1177b790b663281635e15e73f"><td class="memItemLeft" align="right" valign="top">const MatrixType & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1RealQZ.html#a0f5765f1177b790b663281635e15e73f">matrixZ</a> () const </td></tr>
<tr class="memdesc:a0f5765f1177b790b663281635e15e73f"><td class="mdescLeft"> </td><td class="mdescRight">Returns matrix Z in the QZ decomposition. <a href="#a0f5765f1177b790b663281635e15e73f">More...</a><br/></td></tr>
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<tr class="memitem:a4b2119ce39103693d003d8e434f00a3a"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1RealQZ.html#a4b2119ce39103693d003d8e434f00a3a">RealQZ</a> (Index size=RowsAtCompileTime==<a class="el" href="namespaceEigen.html#adc9da5be31bdce40c25a92c27999c0e3">Dynamic</a>?1:RowsAtCompileTime)</td></tr>
<tr class="memdesc:a4b2119ce39103693d003d8e434f00a3a"><td class="mdescLeft"> </td><td class="mdescRight">Default constructor. <a href="#a4b2119ce39103693d003d8e434f00a3a">More...</a><br/></td></tr>
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<tr class="memitem:ab81ec305afdcbf94ed288d382710e8d7"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1RealQZ.html#ab81ec305afdcbf94ed288d382710e8d7">RealQZ</a> (const MatrixType &A, const MatrixType &B, bool computeQZ=true)</td></tr>
<tr class="memdesc:ab81ec305afdcbf94ed288d382710e8d7"><td class="mdescLeft"> </td><td class="mdescRight">Constructor; computes real QZ decomposition of given matrices. <a href="#ab81ec305afdcbf94ed288d382710e8d7">More...</a><br/></td></tr>
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<tr class="memitem:a1b369841b0e39a1ac80a6c32b721d242"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1RealQZ.html">RealQZ</a> & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1RealQZ.html#a1b369841b0e39a1ac80a6c32b721d242">setMaxIterations</a> (Index maxIters)</td></tr>
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<h2 class="groupheader">Constructor & Destructor Documentation</h2>
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<td class="paramtype">Index </td>
<td class="paramname"><em>size</em> = <code>RowsAtCompileTime==<a class="el" href="namespaceEigen.html#adc9da5be31bdce40c25a92c27999c0e3">Dynamic</a> ? 1 : RowsAtCompileTime</code></td><td>)</td>
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<p>Default constructor. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">size</td><td>Positive integer, size of the matrix whose QZ decomposition will be computed.</td></tr>
</table>
</dd>
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<p>The default constructor is useful in cases in which the user intends to perform decompositions via <a class="el" href="classEigen_1_1RealQZ.html#a75cc509be2fa23cde7371b79ebd9f770" title="Computes QZ decomposition of given matrix. ">compute()</a>. The <code>size</code> parameter is only used as a hint. It is not an error to give a wrong <code>size</code>, but it may impair performance.</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1RealQZ.html#a75cc509be2fa23cde7371b79ebd9f770" title="Computes QZ decomposition of given matrix. ">compute()</a> for an example. </dd></dl>
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<p>Constructor; computes real QZ decomposition of given matrices. </p>
<dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramdir">[in]</td><td class="paramname">A</td><td><a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> A. </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">B</td><td><a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> B. </td></tr>
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<p>This constructor calls <a class="el" href="classEigen_1_1RealQZ.html#a75cc509be2fa23cde7371b79ebd9f770" title="Computes QZ decomposition of given matrix. ">compute()</a> to compute the QZ decomposition. </p>
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<h2 class="groupheader">Member Function Documentation</h2>
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<td class="memname"><a class="el" href="classEigen_1_1RealQZ.html">RealQZ</a>< MatrixType > & compute </td>
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<p>Computes QZ decomposition of given matrix. </p>
<dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramdir">[in]</td><td class="paramname">A</td><td><a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> A. </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">B</td><td><a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> B. </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">computeQZ</td><td>If false, A and Z are not computed. </td></tr>
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<dl class="section return"><dt>Returns</dt><dd>Reference to <code>*this</code> </dd></dl>
<p>References <a class="el" href="group__enums.html#gga51bc1ac16f26ebe51eae1abb77bd037ba4ff235bd185f3c5fceeec8d6540eb847">Eigen::NoConvergence</a>, and <a class="el" href="group__enums.html#gga51bc1ac16f26ebe51eae1abb77bd037bafdfbdf3247bd36a1f17270d5cec74c9c">Eigen::Success</a>.</p>
<p>Referenced by <a class="el" href="classEigen_1_1RealQZ.html#ab81ec305afdcbf94ed288d382710e8d7">RealQZ< MatrixType >::RealQZ()</a>.</p>
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<p>Reports whether previous computation was successful. </p>
<dl class="section return"><dt>Returns</dt><dd><code>Success</code> if computation was succesful, <code>NoConvergence</code> otherwise. </dd></dl>
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<p>Returns matrix Q in the QZ decomposition. </p>
<dl class="section return"><dt>Returns</dt><dd>A const reference to the matrix Q. </dd></dl>
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<p>Returns matrix S in the QZ decomposition. </p>
<dl class="section return"><dt>Returns</dt><dd>A const reference to the matrix S. </dd></dl>
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<p>Returns matrix S in the QZ decomposition. </p>
<dl class="section return"><dt>Returns</dt><dd>A const reference to the matrix S. </dd></dl>
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<p>Returns matrix Z in the QZ decomposition. </p>
<dl class="section return"><dt>Returns</dt><dd>A const reference to the matrix Z. </dd></dl>
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<td class="memname"><a class="el" href="classEigen_1_1RealQZ.html">RealQZ</a>& setMaxIterations </td>
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<p>Sets the maximal number of iterations allowed to converge to one eigenvalue or decouple the problem. </p>
<p>Referenced by <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#af6db2eac907628bd50de67a4ede88a5a">GeneralizedEigenSolver< _MatrixType >::setMaxIterations()</a>.</p>
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<hr/>The documentation for this class was generated from the following file:<ul>
<li><a class="el" href="RealQZ_8h_source.html">RealQZ.h</a></li>
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