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<div class="title">GeneralizedEigenSolver< _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::GeneralizedEigenSolver< _MatrixType ></h3>
<p>Computes the generalized eigenvalues and eigenvectors of a pair of general 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 matrices of which we are computing the eigen-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. Currently, only real matrices are supported.</td></tr>
</table>
</dd>
</dl>
<p>The generalized eigenvalues and eigenvectors of a matrix pair <img class="formulaInl" alt="$ A $" src="form_1.png"/> and <img class="formulaInl" alt="$ B $" src="form_56.png"/> are scalars <img class="formulaInl" alt="$ \lambda $" src="form_38.png"/> and vectors <img class="formulaInl" alt="$ v $" src="form_13.png"/> such that <img class="formulaInl" alt="$ Av = \lambda Bv $" src="form_57.png"/>. If <img class="formulaInl" alt="$ D $" src="form_10.png"/> is a diagonal matrix with the eigenvalues on the diagonal, and <img class="formulaInl" alt="$ V $" src="form_40.png"/> is a matrix with the eigenvectors as its columns, then <img class="formulaInl" alt="$ A V = B V D $" src="form_58.png"/>. The matrix <img class="formulaInl" alt="$ V $" src="form_40.png"/> is almost always invertible, in which case we have <img class="formulaInl" alt="$ A = B V D V^{-1} $" src="form_59.png"/>. This is called the generalized eigen-decomposition.</p>
<p>The generalized eigenvalues and eigenvectors of a matrix pair may be complex, even when the matrices are real. Moreover, the generalized eigenvalue might be infinite if the matrix B is singular. To workaround this difficulty, the eigenvalues are provided as a pair of complex <img class="formulaInl" alt="$ \alpha $" src="form_60.png"/> and real <img class="formulaInl" alt="$ \beta $" src="form_61.png"/> such that: <img class="formulaInl" alt="$ \lambda_i = \alpha_i / \beta_i $" src="form_62.png"/>. If <img class="formulaInl" alt="$ \beta_i $" src="form_63.png"/> is (nearly) zero, then one can consider the well defined left eigenvalue <img class="formulaInl" alt="$ \mu = \beta_i / \alpha_i$" src="form_64.png"/> such that: <img class="formulaInl" alt="$ \mu_i A v_i = B v_i $" src="form_65.png"/>, or even <img class="formulaInl" alt="$ \mu_i u_i^T A = u_i^T B $" src="form_66.png"/> where <img class="formulaInl" alt="$ u_i $" src="form_67.png"/> is called the left eigenvector.</p>
<p>Call the function <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a64e32bd0f28bb7b6e91775c3ac592e12" title="Computes generalized eigendecomposition of given matrix. ">compute()</a> to compute the generalized eigenvalues and eigenvectors of a given matrix pair. Alternatively, you can use the <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aa0f561be93f959404ee232321b389468" title="Constructor; computes the generalized eigendecomposition of given matrix pair. ">GeneralizedEigenSolver(const MatrixType&, const MatrixType&, bool)</a> constructor which computes the eigenvalues and eigenvectors at construction time. Once the eigenvalue and eigenvectors are computed, they can be retrieved with the <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#acffd08bee548eaa5c10414343a93529f" title="Returns an expression of the computed generalized eigenvalues. ">eigenvalues()</a> and eigenvectors() functions.</p>
<p>Here is an usage example of this class: Example: </p>
<div class="fragment"><div class="line">GeneralizedEigenSolver<MatrixXf> ges;</div>
<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">ges.compute(A, B);</div>
<div class="line">cout << <span class="stringliteral">"The (complex) numerators of the generalzied eigenvalues are: "</span> << ges.alphas().transpose() << endl;</div>
<div class="line">cout << <span class="stringliteral">"The (real) denominatore of the generalzied eigenvalues are: "</span> << ges.betas().transpose() << endl;</div>
<div class="line">cout << <span class="stringliteral">"The (complex) generalzied eigenvalues are (alphas./beta): "</span> << ges.eigenvalues().transpose() << endl;</div>
</div><!-- fragment --><p> Output: </p>
<pre class="fragment">The (complex) numerators of the generalzied eigenvalues are: (0.644,0.795) (0.644,-0.795) (-0.398,0) (-1.12,0)
The (real) denominatore of the generalzied eigenvalues are: 1.51 1.51 -1.25 0.746
The (complex) generalzied eigenvalues are (alphas./beta): (0.427,0.528) (0.427,-0.528) (0.318,-0) (-1.5,0)
</pre><dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1MatrixBase.html#a0ffa061371b1bd1b9f14ecef94b4502e" title="Computes the eigenvalues of a matrix. ">MatrixBase::eigenvalues()</a>, class <a class="el" href="classEigen_1_1ComplexEigenSolver.html" title="Computes eigenvalues and eigenvectors of general complex matrices. ">ComplexEigenSolver</a>, class <a class="el" href="classEigen_1_1SelfAdjointEigenSolver.html" title="Computes eigenvalues and eigenvectors of selfadjoint matrices. ">SelfAdjointEigenSolver</a> </dd></dl>
</div><table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="pub-types"></a>
Public Types</h2></td></tr>
<tr class="memitem:a1b9bc0a45616064df3a6168395e3cfcc"><td class="memItemLeft" align="right" valign="top">typedef std::complex< RealScalar > </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a1b9bc0a45616064df3a6168395e3cfcc">ComplexScalar</a></td></tr>
<tr class="memdesc:a1b9bc0a45616064df3a6168395e3cfcc"><td class="mdescLeft"> </td><td class="mdescRight">Complex scalar type for <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aeb6c0eb89cc982629305f6c7e0791caf" title="Synonym for the template parameter _MatrixType. ">MatrixType</a>. <a href="#a1b9bc0a45616064df3a6168395e3cfcc">More...</a><br/></td></tr>
<tr class="separator:a1b9bc0a45616064df3a6168395e3cfcc"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a12de5e55557c63e5efaa70c3d4e82060"><td class="memItemLeft" align="right" valign="top">typedef <a class="el" href="classEigen_1_1Matrix.html">Matrix</a>< <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a1b9bc0a45616064df3a6168395e3cfcc">ComplexScalar</a>, <br class="typebreak"/>
ColsAtCompileTime, 1, Options <br class="typebreak"/>
&~<a class="el" href="group__enums.html#gga0c5bde183ecefe103f70b49ad9740bcda1e16fa1b92ed7a058cd4ce7a9a0db044">RowMajor</a>, <br class="typebreak"/>
MaxColsAtCompileTime, 1 > </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a12de5e55557c63e5efaa70c3d4e82060">ComplexVectorType</a></td></tr>
<tr class="memdesc:a12de5e55557c63e5efaa70c3d4e82060"><td class="mdescLeft"> </td><td class="mdescRight">Type for vector of complex scalar values eigenvalues as returned by <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a92c089793f883fd38cd2df4d7c6513e5">betas()</a>. <a href="#a12de5e55557c63e5efaa70c3d4e82060">More...</a><br/></td></tr>
<tr class="separator:a12de5e55557c63e5efaa70c3d4e82060"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a14f2d5bf9df5a70ea27bc1239aa30822"><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a14f2d5bf9df5a70ea27bc1239aa30822"></a>
typedef <a class="el" href="classEigen_1_1CwiseBinaryOp.html">CwiseBinaryOp</a><br class="typebreak"/>
< internal::scalar_quotient_op<br class="typebreak"/>
< <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a1b9bc0a45616064df3a6168395e3cfcc">ComplexScalar</a>, <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a3f6fc00047c205ee590f676934aab28f">Scalar</a> ><br class="typebreak"/>
, <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a12de5e55557c63e5efaa70c3d4e82060">ComplexVectorType</a>, <br class="typebreak"/>
<a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#ae88654b9613217486067f07e394c88dc">VectorType</a> > </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a14f2d5bf9df5a70ea27bc1239aa30822">EigenvalueType</a></td></tr>
<tr class="memdesc:a14f2d5bf9df5a70ea27bc1239aa30822"><td class="mdescLeft"> </td><td class="mdescRight">Expression type for the eigenvalues as returned by <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#acffd08bee548eaa5c10414343a93529f" title="Returns an expression of the computed generalized eigenvalues. ">eigenvalues()</a>. <br/></td></tr>
<tr class="separator:a14f2d5bf9df5a70ea27bc1239aa30822"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a50d070013a795db5621119f2b4a3d781"><td class="memItemLeft" align="right" valign="top">typedef <a class="el" href="classEigen_1_1Matrix.html">Matrix</a>< <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a1b9bc0a45616064df3a6168395e3cfcc">ComplexScalar</a>, <br class="typebreak"/>
RowsAtCompileTime, <br class="typebreak"/>
ColsAtCompileTime, Options, <br class="typebreak"/>
MaxRowsAtCompileTime, <br class="typebreak"/>
MaxColsAtCompileTime > </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a50d070013a795db5621119f2b4a3d781">EigenvectorsType</a></td></tr>
<tr class="memdesc:a50d070013a795db5621119f2b4a3d781"><td class="mdescLeft"> </td><td class="mdescRight">Type for matrix of eigenvectors as returned by eigenvectors(). <a href="#a50d070013a795db5621119f2b4a3d781">More...</a><br/></td></tr>
<tr class="separator:a50d070013a795db5621119f2b4a3d781"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aeb6c0eb89cc982629305f6c7e0791caf"><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="aeb6c0eb89cc982629305f6c7e0791caf"></a>
typedef _MatrixType </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a></td></tr>
<tr class="memdesc:aeb6c0eb89cc982629305f6c7e0791caf"><td class="mdescLeft"> </td><td class="mdescRight">Synonym for the template parameter <code>_MatrixType</code>. <br/></td></tr>
<tr class="separator:aeb6c0eb89cc982629305f6c7e0791caf"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a3f6fc00047c205ee590f676934aab28f"><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a3f6fc00047c205ee590f676934aab28f"></a>
typedef MatrixType::Scalar </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a3f6fc00047c205ee590f676934aab28f">Scalar</a></td></tr>
<tr class="memdesc:a3f6fc00047c205ee590f676934aab28f"><td class="mdescLeft"> </td><td class="mdescRight">Scalar type for matrices of type <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aeb6c0eb89cc982629305f6c7e0791caf" title="Synonym for the template parameter _MatrixType. ">MatrixType</a>. <br/></td></tr>
<tr class="separator:a3f6fc00047c205ee590f676934aab28f"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ae88654b9613217486067f07e394c88dc"><td class="memItemLeft" align="right" valign="top">typedef <a class="el" href="classEigen_1_1Matrix.html">Matrix</a>< <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a3f6fc00047c205ee590f676934aab28f">Scalar</a>, <br class="typebreak"/>
ColsAtCompileTime, 1, Options <br class="typebreak"/>
&~<a class="el" href="group__enums.html#gga0c5bde183ecefe103f70b49ad9740bcda1e16fa1b92ed7a058cd4ce7a9a0db044">RowMajor</a>, <br class="typebreak"/>
MaxColsAtCompileTime, 1 > </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#ae88654b9613217486067f07e394c88dc">VectorType</a></td></tr>
<tr class="memdesc:ae88654b9613217486067f07e394c88dc"><td class="mdescLeft"> </td><td class="mdescRight">Type for vector of real scalar values eigenvalues as returned by <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a92c089793f883fd38cd2df4d7c6513e5">betas()</a>. <a href="#ae88654b9613217486067f07e394c88dc">More...</a><br/></td></tr>
<tr class="separator:ae88654b9613217486067f07e394c88dc"><td class="memSeparator" colspan="2"> </td></tr>
</table><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:ade79c282ebff0a23829fd9b9b18fe1c0"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a12de5e55557c63e5efaa70c3d4e82060">ComplexVectorType</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#ade79c282ebff0a23829fd9b9b18fe1c0">alphas</a> () const </td></tr>
<tr class="separator:ade79c282ebff0a23829fd9b9b18fe1c0"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a92c089793f883fd38cd2df4d7c6513e5"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#ae88654b9613217486067f07e394c88dc">VectorType</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a92c089793f883fd38cd2df4d7c6513e5">betas</a> () const </td></tr>
<tr class="separator:a92c089793f883fd38cd2df4d7c6513e5"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a64e32bd0f28bb7b6e91775c3ac592e12"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html">GeneralizedEigenSolver</a> & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a64e32bd0f28bb7b6e91775c3ac592e12">compute</a> (const <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a> &A, const <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a> &B, bool computeEigenvectors=true)</td></tr>
<tr class="memdesc:a64e32bd0f28bb7b6e91775c3ac592e12"><td class="mdescLeft"> </td><td class="mdescRight">Computes generalized eigendecomposition of given matrix. <a href="#a64e32bd0f28bb7b6e91775c3ac592e12">More...</a><br/></td></tr>
<tr class="separator:a64e32bd0f28bb7b6e91775c3ac592e12"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:acffd08bee548eaa5c10414343a93529f"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a14f2d5bf9df5a70ea27bc1239aa30822">EigenvalueType</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#acffd08bee548eaa5c10414343a93529f">eigenvalues</a> () const </td></tr>
<tr class="memdesc:acffd08bee548eaa5c10414343a93529f"><td class="mdescLeft"> </td><td class="mdescRight">Returns an expression of the computed generalized eigenvalues. <a href="#acffd08bee548eaa5c10414343a93529f">More...</a><br/></td></tr>
<tr class="separator:acffd08bee548eaa5c10414343a93529f"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a7289bdfc65bb36b51babcab21a1a9c14"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a7289bdfc65bb36b51babcab21a1a9c14">GeneralizedEigenSolver</a> ()</td></tr>
<tr class="memdesc:a7289bdfc65bb36b51babcab21a1a9c14"><td class="mdescLeft"> </td><td class="mdescRight">Default constructor. <a href="#a7289bdfc65bb36b51babcab21a1a9c14">More...</a><br/></td></tr>
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<tr class="memdesc:ad454c37be646409992625586d2b21adb"><td class="mdescLeft"> </td><td class="mdescRight">Default constructor with memory preallocation. <a href="#ad454c37be646409992625586d2b21adb">More...</a><br/></td></tr>
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<tr class="memitem:aa0f561be93f959404ee232321b389468"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aa0f561be93f959404ee232321b389468">GeneralizedEigenSolver</a> (const <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a> &A, const <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a> &B, bool computeEigenvectors=true)</td></tr>
<tr class="memdesc:aa0f561be93f959404ee232321b389468"><td class="mdescLeft"> </td><td class="mdescRight">Constructor; computes the generalized eigendecomposition of given matrix pair. <a href="#aa0f561be93f959404ee232321b389468">More...</a><br/></td></tr>
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<h2 class="groupheader">Member Typedef Documentation</h2>
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<td class="memname">typedef std::complex<RealScalar> <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a1b9bc0a45616064df3a6168395e3cfcc">ComplexScalar</a></td>
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<p>Complex scalar type for <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aeb6c0eb89cc982629305f6c7e0791caf" title="Synonym for the template parameter _MatrixType. ">MatrixType</a>. </p>
<p>This is <code>std::complex<Scalar></code> if <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a3f6fc00047c205ee590f676934aab28f" title="Scalar type for matrices of type MatrixType. ">Scalar</a> is real (e.g., <code>float</code> or <code>double</code>) and just <code>Scalar</code> if <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a3f6fc00047c205ee590f676934aab28f" title="Scalar type for matrices of type MatrixType. ">Scalar</a> is complex. </p>
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<td class="memname">typedef <a class="el" href="classEigen_1_1Matrix.html">Matrix</a><<a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a1b9bc0a45616064df3a6168395e3cfcc">ComplexScalar</a>, ColsAtCompileTime, 1, Options & ~<a class="el" href="group__enums.html#gga0c5bde183ecefe103f70b49ad9740bcda1e16fa1b92ed7a058cd4ce7a9a0db044">RowMajor</a>, MaxColsAtCompileTime, 1> <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a12de5e55557c63e5efaa70c3d4e82060">ComplexVectorType</a></td>
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<p>Type for vector of complex scalar values eigenvalues as returned by <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a92c089793f883fd38cd2df4d7c6513e5">betas()</a>. </p>
<p>This is a column vector with entries of type <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a1b9bc0a45616064df3a6168395e3cfcc" title="Complex scalar type for MatrixType. ">ComplexScalar</a>. The length of the vector is the size of <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aeb6c0eb89cc982629305f6c7e0791caf" title="Synonym for the template parameter _MatrixType. ">MatrixType</a>. </p>
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<td class="memname">typedef <a class="el" href="classEigen_1_1Matrix.html">Matrix</a><<a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a1b9bc0a45616064df3a6168395e3cfcc">ComplexScalar</a>, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a50d070013a795db5621119f2b4a3d781">EigenvectorsType</a></td>
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<p>Type for matrix of eigenvectors as returned by eigenvectors(). </p>
<p>This is a square matrix with entries of type <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a1b9bc0a45616064df3a6168395e3cfcc" title="Complex scalar type for MatrixType. ">ComplexScalar</a>. The size is the same as the size of <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aeb6c0eb89cc982629305f6c7e0791caf" title="Synonym for the template parameter _MatrixType. ">MatrixType</a>. </p>
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<td class="memname">typedef <a class="el" href="classEigen_1_1Matrix.html">Matrix</a><<a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a3f6fc00047c205ee590f676934aab28f">Scalar</a>, ColsAtCompileTime, 1, Options & ~<a class="el" href="group__enums.html#gga0c5bde183ecefe103f70b49ad9740bcda1e16fa1b92ed7a058cd4ce7a9a0db044">RowMajor</a>, MaxColsAtCompileTime, 1> <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#ae88654b9613217486067f07e394c88dc">VectorType</a></td>
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<p>Type for vector of real scalar values eigenvalues as returned by <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a92c089793f883fd38cd2df4d7c6513e5">betas()</a>. </p>
<p>This is a column vector with entries of type <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a3f6fc00047c205ee590f676934aab28f" title="Scalar type for matrices of type MatrixType. ">Scalar</a>. The length of the vector is the size of <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aeb6c0eb89cc982629305f6c7e0791caf" title="Synonym for the template parameter _MatrixType. ">MatrixType</a>. </p>
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<h2 class="groupheader">Constructor & Destructor Documentation</h2>
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<td class="memname"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html">GeneralizedEigenSolver</a> </td>
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<p>Default constructor. </p>
<p>The default constructor is useful in cases in which the user intends to perform decompositions via <a class="el" href="classEigen_1_1EigenSolver.html#a0e257dae8f1774fdda178482caa65be8" title="Computes eigendecomposition of given matrix. ">EigenSolver::compute(const MatrixType&, bool)</a>.</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a64e32bd0f28bb7b6e91775c3ac592e12" title="Computes generalized eigendecomposition of given matrix. ">compute()</a> for an example. </dd></dl>
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<td class="memname"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html">GeneralizedEigenSolver</a> </td>
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<td class="paramname"><em>size</em></td><td>)</td>
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<p>Default constructor with memory preallocation. </p>
<p>Like the default constructor but with preallocation of the internal data according to the specified problem <em>size</em>. </p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a7289bdfc65bb36b51babcab21a1a9c14" title="Default constructor. ">GeneralizedEigenSolver()</a> </dd></dl>
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<td class="memname"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html">GeneralizedEigenSolver</a> </td>
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<td class="paramtype">const <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a> & </td>
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<p>Constructor; computes the generalized eigendecomposition of given matrix pair. </p>
<dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramdir">[in]</td><td class="paramname">A</td><td>Square matrix whose eigendecomposition is to be computed. </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">B</td><td>Square matrix whose eigendecomposition is to be computed. </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">computeEigenvectors</td><td>If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.</td></tr>
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<p>This constructor calls <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a64e32bd0f28bb7b6e91775c3ac592e12" title="Computes generalized eigendecomposition of given matrix. ">compute()</a> to compute the generalized eigenvalues and eigenvectors.</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a64e32bd0f28bb7b6e91775c3ac592e12" title="Computes generalized eigendecomposition of given matrix. ">compute()</a> </dd></dl>
<p>References <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a64e32bd0f28bb7b6e91775c3ac592e12">GeneralizedEigenSolver< _MatrixType >::compute()</a>.</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_1GeneralizedEigenSolver.html#a12de5e55557c63e5efaa70c3d4e82060">ComplexVectorType</a> alphas </td>
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<dl class="section return"><dt>Returns</dt><dd>A const reference to the vectors containing the alpha values</dd></dl>
<p>This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a92c089793f883fd38cd2df4d7c6513e5">betas()</a>, <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#acffd08bee548eaa5c10414343a93529f" title="Returns an expression of the computed generalized eigenvalues. ">eigenvalues()</a> </dd></dl>
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<td class="memname"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#ae88654b9613217486067f07e394c88dc">VectorType</a> betas </td>
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<dl class="section return"><dt>Returns</dt><dd>A const reference to the vectors containing the beta values</dd></dl>
<p>This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#ade79c282ebff0a23829fd9b9b18fe1c0">alphas()</a>, <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#acffd08bee548eaa5c10414343a93529f" title="Returns an expression of the computed generalized eigenvalues. ">eigenvalues()</a> </dd></dl>
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<td class="memname"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html">GeneralizedEigenSolver</a>< <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a> > & compute </td>
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<p>Computes generalized eigendecomposition 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>Square matrix whose eigendecomposition is to be computed. </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">B</td><td>Square matrix whose eigendecomposition is to be computed. </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">computeEigenvectors</td><td>If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed. </td></tr>
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<dl class="section return"><dt>Returns</dt><dd>Reference to <code>*this</code> </dd></dl>
<p>This function computes the eigenvalues of the real matrix <code>matrix</code>. The <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#acffd08bee548eaa5c10414343a93529f" title="Returns an expression of the computed generalized eigenvalues. ">eigenvalues()</a> function can be used to retrieve them. If <code>computeEigenvectors</code> is true, then the eigenvectors are also computed and can be retrieved by calling eigenvectors().</p>
<p>The matrix is first reduced to real generalized Schur form using the <a class="el" href="classEigen_1_1RealQZ.html" title="Performs a real QZ decomposition of a pair of square matrices. ">RealQZ</a> class. The generalized Schur decomposition is then used to compute the eigenvalues and eigenvectors.</p>
<p>The cost of the computation is dominated by the cost of the generalized Schur decomposition.</p>
<p>This method reuses of the allocated data in the <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html" title="Computes the generalized eigenvalues and eigenvectors of a pair of general matrices. ">GeneralizedEigenSolver</a> object. </p>
<p>References <a class="el" href="group__enums.html#gga51bc1ac16f26ebe51eae1abb77bd037bafdfbdf3247bd36a1f17270d5cec74c9c">Eigen::Success</a>.</p>
<p>Referenced by <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aa0f561be93f959404ee232321b389468">GeneralizedEigenSolver< _MatrixType >::GeneralizedEigenSolver()</a>.</p>
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<td class="memname"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a14f2d5bf9df5a70ea27bc1239aa30822">EigenvalueType</a> eigenvalues </td>
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<p>Returns an expression of the computed generalized eigenvalues. </p>
<dl class="section return"><dt>Returns</dt><dd>An expression of the column vector containing the eigenvalues.</dd></dl>
<p>It is a shortcut for</p>
<div class="fragment"><div class="line">this-><a class="code" href="classEigen_1_1GeneralizedEigenSolver.html#ade79c282ebff0a23829fd9b9b18fe1c0">alphas</a>().cwiseQuotient(this-><a class="code" href="classEigen_1_1GeneralizedEigenSolver.html#a92c089793f883fd38cd2df4d7c6513e5">betas</a>()); </div>
</div><!-- fragment --><p> Not that betas might contain zeros. It is therefore not recommended to use this function, but rather directly deal with the alphas and betas vectors.</p>
<dl class="section pre"><dt>Precondition</dt><dd>Either the constructor <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#aa0f561be93f959404ee232321b389468" title="Constructor; computes the generalized eigendecomposition of given matrix pair. ">GeneralizedEigenSolver(const MatrixType&,const MatrixType&,bool)</a> or the member function <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a64e32bd0f28bb7b6e91775c3ac592e12" title="Computes generalized eigendecomposition of given matrix. ">compute(const MatrixType&,const MatrixType&,bool)</a> has been called before.</dd></dl>
<p>The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are not sorted in any particular order.</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#ade79c282ebff0a23829fd9b9b18fe1c0">alphas()</a>, <a class="el" href="classEigen_1_1GeneralizedEigenSolver.html#a92c089793f883fd38cd2df4d7c6513e5">betas()</a>, eigenvectors() </dd></dl>
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<td class="memname"><a class="el" href="classEigen_1_1GeneralizedEigenSolver.html">GeneralizedEigenSolver</a>& setMaxIterations </td>
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<p>Sets the maximal number of iterations allowed. </p>
<p>References <a class="el" href="classEigen_1_1RealQZ.html#a1b369841b0e39a1ac80a6c32b721d242">RealQZ< _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="GeneralizedEigenSolver_8h_source.html">GeneralizedEigenSolver.h</a></li>
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