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<div class="title">Tridiagonalization< _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::Tridiagonalization< _MatrixType ></h3>
<p>Tridiagonal decomposition of a selfadjoint matrix. </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 tridiagonal 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>This class performs a tridiagonal decomposition of a selfadjoint matrix <img class="formulaInl" alt="$ A $" src="form_1.png"/> such that: <img class="formulaInl" alt="$ A = Q T Q^* $" src="form_98.png"/> where <img class="formulaInl" alt="$ Q $" src="form_73.png"/> is unitary and <img class="formulaInl" alt="$ T $" src="form_99.png"/> a real symmetric tridiagonal matrix.</p>
<p>A tridiagonal matrix is a matrix which has nonzero elements only on the main diagonal and the first diagonal below and above it. The Hessenberg decomposition of a selfadjoint matrix is in fact a tridiagonal decomposition. This class is used in <a class="el" href="classEigen_1_1SelfAdjointEigenSolver.html" title="Computes eigenvalues and eigenvectors of selfadjoint matrices. ">SelfAdjointEigenSolver</a> to compute the eigenvalues and eigenvectors of a selfadjoint matrix.</p>
<p>Call the function <a class="el" href="classEigen_1_1Tridiagonalization.html#aa69e607a4aab4fb6321ca6acbf074fc2" title="Computes tridiagonal decomposition of given matrix. ">compute()</a> to compute the tridiagonal decomposition of a given matrix. Alternatively, you can use the <a class="el" href="classEigen_1_1Tridiagonalization.html#aa9f9722d2cef9425e2c0da3553dfbac7" title="Constructor; computes tridiagonal decomposition of given matrix. ">Tridiagonalization(const MatrixType&)</a> constructor which computes the tridiagonal Schur decomposition at construction time. Once the decomposition is computed, you can use the <a class="el" href="classEigen_1_1Tridiagonalization.html#ad13845d7490115664924b3dc208ec369" title="Returns the unitary matrix Q in the decomposition. ">matrixQ()</a> and <a class="el" href="classEigen_1_1Tridiagonalization.html#aceb0f16a166f4c236a1b536b7424d292" title="Returns an expression of the tridiagonal matrix T in the decomposition. ">matrixT()</a> functions to retrieve the matrices Q and T in the decomposition.</p>
<p>The documentation of <a class="el" href="classEigen_1_1Tridiagonalization.html#aa9f9722d2cef9425e2c0da3553dfbac7" title="Constructor; computes tridiagonal decomposition of given matrix. ">Tridiagonalization(const MatrixType&)</a> contains an example of the typical use of this class.</p>
<dl class="section see"><dt>See Also</dt><dd>class <a class="el" href="classEigen_1_1HessenbergDecomposition.html" title="Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation. ">HessenbergDecomposition</a>, class <a class="el" href="classEigen_1_1SelfAdjointEigenSolver.html" title="Computes eigenvalues and eigenvectors of selfadjoint matrices. ">SelfAdjointEigenSolver</a> </dd></dl>
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Public Types</h2></td></tr>
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typedef <a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequence</a><br class="typebreak"/>
< <a class="el" href="classEigen_1_1Tridiagonalization.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a>, typename <br class="typebreak"/>
internal::remove_all< typename <br class="typebreak"/>
CoeffVectorType::ConjugateReturnType ><br class="typebreak"/>
::type > </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Tridiagonalization.html#aa96bdbc1b19c647e3372c31301ea4999">HouseholderSequenceType</a></td></tr>
<tr class="memdesc:aa96bdbc1b19c647e3372c31301ea4999"><td class="mdescLeft"> </td><td class="mdescRight">Return type of <a class="el" href="classEigen_1_1Tridiagonalization.html#ad13845d7490115664924b3dc208ec369" title="Returns the unitary matrix Q in the decomposition. ">matrixQ()</a> <br/></td></tr>
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<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_1Tridiagonalization.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>
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<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="pub-methods"></a>
Public Member Functions</h2></td></tr>
<tr class="memitem:aa69e607a4aab4fb6321ca6acbf074fc2"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1Tridiagonalization.html">Tridiagonalization</a> & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Tridiagonalization.html#aa69e607a4aab4fb6321ca6acbf074fc2">compute</a> (const <a class="el" href="classEigen_1_1Tridiagonalization.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a> &matrix)</td></tr>
<tr class="memdesc:aa69e607a4aab4fb6321ca6acbf074fc2"><td class="mdescLeft"> </td><td class="mdescRight">Computes tridiagonal decomposition of given matrix. <a href="#aa69e607a4aab4fb6321ca6acbf074fc2">More...</a><br/></td></tr>
<tr class="separator:aa69e607a4aab4fb6321ca6acbf074fc2"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ac109eefddd733d8e82841da5bb2dd8d3"><td class="memItemLeft" align="right" valign="top">DiagonalReturnType </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Tridiagonalization.html#ac109eefddd733d8e82841da5bb2dd8d3">diagonal</a> () const </td></tr>
<tr class="memdesc:ac109eefddd733d8e82841da5bb2dd8d3"><td class="mdescLeft"> </td><td class="mdescRight">Returns the diagonal of the tridiagonal matrix T in the decomposition. <a href="#ac109eefddd733d8e82841da5bb2dd8d3">More...</a><br/></td></tr>
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<tr class="memitem:a2ab889c75460c178d941ee24e371b206"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1Matrix.html">CoeffVectorType</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Tridiagonalization.html#a2ab889c75460c178d941ee24e371b206">householderCoefficients</a> () const </td></tr>
<tr class="memdesc:a2ab889c75460c178d941ee24e371b206"><td class="mdescLeft"> </td><td class="mdescRight">Returns the Householder coefficients. <a href="#a2ab889c75460c178d941ee24e371b206">More...</a><br/></td></tr>
<tr class="separator:a2ab889c75460c178d941ee24e371b206"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ad13845d7490115664924b3dc208ec369"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1Tridiagonalization.html#aa96bdbc1b19c647e3372c31301ea4999">HouseholderSequenceType</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Tridiagonalization.html#ad13845d7490115664924b3dc208ec369">matrixQ</a> () const </td></tr>
<tr class="memdesc:ad13845d7490115664924b3dc208ec369"><td class="mdescLeft"> </td><td class="mdescRight">Returns the unitary matrix Q in the decomposition. <a href="#ad13845d7490115664924b3dc208ec369">More...</a><br/></td></tr>
<tr class="separator:ad13845d7490115664924b3dc208ec369"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aceb0f16a166f4c236a1b536b7424d292"><td class="memItemLeft" align="right" valign="top">MatrixTReturnType </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Tridiagonalization.html#aceb0f16a166f4c236a1b536b7424d292">matrixT</a> () const </td></tr>
<tr class="memdesc:aceb0f16a166f4c236a1b536b7424d292"><td class="mdescLeft"> </td><td class="mdescRight">Returns an expression of the tridiagonal matrix T in the decomposition. <a href="#aceb0f16a166f4c236a1b536b7424d292">More...</a><br/></td></tr>
<tr class="separator:aceb0f16a166f4c236a1b536b7424d292"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a66adece364b64b26b3771662de70f2df"><td class="memItemLeft" align="right" valign="top">const <a class="el" href="classEigen_1_1Tridiagonalization.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a> & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Tridiagonalization.html#a66adece364b64b26b3771662de70f2df">packedMatrix</a> () const </td></tr>
<tr class="memdesc:a66adece364b64b26b3771662de70f2df"><td class="mdescLeft"> </td><td class="mdescRight">Returns the internal representation of the decomposition. <a href="#a66adece364b64b26b3771662de70f2df">More...</a><br/></td></tr>
<tr class="separator:a66adece364b64b26b3771662de70f2df"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a8fa49216273ab7579b7bea06debb1e51"><td class="memItemLeft" align="right" valign="top">SubDiagonalReturnType </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Tridiagonalization.html#a8fa49216273ab7579b7bea06debb1e51">subDiagonal</a> () const </td></tr>
<tr class="memdesc:a8fa49216273ab7579b7bea06debb1e51"><td class="mdescLeft"> </td><td class="mdescRight">Returns the subdiagonal of the tridiagonal matrix T in the decomposition. <a href="#a8fa49216273ab7579b7bea06debb1e51">More...</a><br/></td></tr>
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<tr class="memitem:a0698ae78b0ab6f239c475b73b9c6bbee"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Tridiagonalization.html#a0698ae78b0ab6f239c475b73b9c6bbee">Tridiagonalization</a> (Index size=Size==<a class="el" href="namespaceEigen.html#adc9da5be31bdce40c25a92c27999c0e3">Dynamic</a>?2:Size)</td></tr>
<tr class="memdesc:a0698ae78b0ab6f239c475b73b9c6bbee"><td class="mdescLeft"> </td><td class="mdescRight">Default constructor. <a href="#a0698ae78b0ab6f239c475b73b9c6bbee">More...</a><br/></td></tr>
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<tr class="memitem:aa9f9722d2cef9425e2c0da3553dfbac7"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1Tridiagonalization.html#aa9f9722d2cef9425e2c0da3553dfbac7">Tridiagonalization</a> (const <a class="el" href="classEigen_1_1Tridiagonalization.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a> &matrix)</td></tr>
<tr class="memdesc:aa9f9722d2cef9425e2c0da3553dfbac7"><td class="mdescLeft"> </td><td class="mdescRight">Constructor; computes tridiagonal decomposition of given matrix. <a href="#aa9f9722d2cef9425e2c0da3553dfbac7">More...</a><br/></td></tr>
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<h2 class="groupheader">Constructor & Destructor Documentation</h2>
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<td>(</td>
<td class="paramtype">Index </td>
<td class="paramname"><em>size</em> = <code>Size==<a class="el" href="namespaceEigen.html#adc9da5be31bdce40c25a92c27999c0e3">Dynamic</a> ? 2 : Size</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 tridiagonal decomposition will be computed.</td></tr>
</table>
</dd>
</dl>
<p>The default constructor is useful in cases in which the user intends to perform decompositions via <a class="el" href="classEigen_1_1Tridiagonalization.html#aa69e607a4aab4fb6321ca6acbf074fc2" title="Computes tridiagonal 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_1Tridiagonalization.html#aa69e607a4aab4fb6321ca6acbf074fc2" title="Computes tridiagonal decomposition of given matrix. ">compute()</a> for an example. </dd></dl>
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<td>(</td>
<td class="paramtype">const <a class="el" href="classEigen_1_1Tridiagonalization.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a> & </td>
<td class="paramname"><em>matrix</em></td><td>)</td>
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<p>Constructor; computes tridiagonal decomposition of given matrix. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">matrix</td><td>Selfadjoint matrix whose tridiagonal decomposition is to be computed.</td></tr>
</table>
</dd>
</dl>
<p>This constructor calls <a class="el" href="classEigen_1_1Tridiagonalization.html#aa69e607a4aab4fb6321ca6acbf074fc2" title="Computes tridiagonal decomposition of given matrix. ">compute()</a> to compute the tridiagonal decomposition.</p>
<p>Example: </p>
<div class="fragment"><div class="line">MatrixXd X = <a class="code" href="classEigen_1_1DenseBase.html#a8e759dafdd9ecc446d397b7f5435f60a">MatrixXd::Random</a>(5,5);</div>
<div class="line">MatrixXd A = X + X.transpose();</div>
<div class="line">cout << <span class="stringliteral">"Here is a random symmetric 5x5 matrix:"</span> << endl << A << endl << endl;</div>
<div class="line">Tridiagonalization<MatrixXd> triOfA(A);</div>
<div class="line">MatrixXd Q = triOfA.matrixQ();</div>
<div class="line">cout << <span class="stringliteral">"The orthogonal matrix Q is:"</span> << endl << Q << endl;</div>
<div class="line">MatrixXd T = triOfA.matrixT();</div>
<div class="line">cout << <span class="stringliteral">"The tridiagonal matrix T is:"</span> << endl << T << endl << endl;</div>
<div class="line">cout << <span class="stringliteral">"Q * T * Q^T = "</span> << endl << Q * T * Q.transpose() << endl;</div>
</div><!-- fragment --><p> Output: </p>
<pre class="fragment">Here is a random symmetric 5x5 matrix:
1.36 -0.816 0.521 1.43 -0.144
-0.816 -0.659 0.794 -0.173 -0.406
0.521 0.794 -0.541 0.461 0.179
1.43 -0.173 0.461 -1.43 0.822
-0.144 -0.406 0.179 0.822 -1.37
The orthogonal matrix Q is:
1 0 0 0 0
0 -0.471 0.127 -0.671 -0.558
0 0.301 -0.195 0.437 -0.825
0 0.825 0.0459 -0.563 -0.00872
0 -0.0832 -0.971 -0.202 0.0922
The tridiagonal matrix T is:
1.36 1.73 0 0 0
1.73 -1.2 -0.966 0 0
0 -0.966 -1.28 0.214 0
0 0 0.214 -1.69 0.345
0 0 0 0.345 0.164
Q * T * Q^T =
1.36 -0.816 0.521 1.43 -0.144
-0.816 -0.659 0.794 -0.173 -0.406
0.521 0.794 -0.541 0.461 0.179
1.43 -0.173 0.461 -1.43 0.822
-0.144 -0.406 0.179 0.822 -1.37
</pre>
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<h2 class="groupheader">Member Function Documentation</h2>
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<td class="memname"><a class="el" href="classEigen_1_1Tridiagonalization.html">Tridiagonalization</a>& compute </td>
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<td class="paramtype">const <a class="el" href="classEigen_1_1Tridiagonalization.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a> & </td>
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<p>Computes tridiagonal 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">matrix</td><td>Selfadjoint matrix whose tridiagonal decomposition is to be computed. </td></tr>
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</dd>
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<dl class="section return"><dt>Returns</dt><dd>Reference to <code>*this</code> </dd></dl>
<p>The tridiagonal decomposition is computed by bringing the columns of the matrix successively in the required form using Householder reflections. The cost is <img class="formulaInl" alt="$ 4n^3/3 $" src="form_95.png"/> flops, where <img class="formulaInl" alt="$ n $" src="form_45.png"/> denotes the size of the given matrix.</p>
<p>This method reuses of the allocated data in the <a class="el" href="classEigen_1_1Tridiagonalization.html" title="Tridiagonal decomposition of a selfadjoint matrix. ">Tridiagonalization</a> object, if the size of the matrix does not change.</p>
<p>Example: </p>
<div class="fragment"><div class="line">Tridiagonalization<MatrixXf> tri;</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#gabab09c32e96cfa9829a88400627af162">MatrixXf</a> X = <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> A = X + X.transpose();</div>
<div class="line">tri.compute(A);</div>
<div class="line">cout << <span class="stringliteral">"The matrix T in the tridiagonal decomposition of A is: "</span> << endl;</div>
<div class="line">cout << tri.matrixT() << endl;</div>
<div class="line">tri.compute(2*A); <span class="comment">// re-use tri to compute eigenvalues of 2A</span></div>
<div class="line">cout << <span class="stringliteral">"The matrix T in the tridiagonal decomposition of 2A is: "</span> << endl;</div>
<div class="line">cout << tri.matrixT() << endl;</div>
</div><!-- fragment --><p> Output: </p>
<pre class="fragment">The matrix T in the tridiagonal decomposition of A is:
1.36 -0.704 0 0
-0.704 0.0147 1.71 0
0 1.71 0.856 0.641
0 0 0.641 -0.506
The matrix T in the tridiagonal decomposition of 2A is:
2.72 -1.41 0 0
-1.41 0.0294 3.43 0
0 3.43 1.71 1.28
0 0 1.28 -1.01
</pre>
<p>References <a class="el" href="classEigen_1_1PlainObjectBase.html#afbbb33d14fe7fb9683019a39ce1c659d">PlainObjectBase< Derived >::resize()</a>.</p>
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<p>Returns the diagonal of the tridiagonal matrix T in the decomposition. </p>
<dl class="section return"><dt>Returns</dt><dd>expression representing the diagonal of T</dd></dl>
<dl class="section pre"><dt>Precondition</dt><dd>Either the constructor <a class="el" href="classEigen_1_1Tridiagonalization.html#aa9f9722d2cef9425e2c0da3553dfbac7" title="Constructor; computes tridiagonal decomposition of given matrix. ">Tridiagonalization(const MatrixType&)</a> or the member function <a class="el" href="classEigen_1_1Tridiagonalization.html#aa69e607a4aab4fb6321ca6acbf074fc2" title="Computes tridiagonal decomposition of given matrix. ">compute(const MatrixType&)</a> has been called before to compute the tridiagonal decomposition of a matrix.</dd></dl>
<p>Example: </p>
<div class="fragment"><div class="line"><a class="code" href="group__matrixtypedefs.html#gaf1d15c8c24df228ee4869535dcbfa288">MatrixXcd</a> X = <a class="code" href="classEigen_1_1DenseBase.html#a8e759dafdd9ecc446d397b7f5435f60a">MatrixXcd::Random</a>(4,4);</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#gaf1d15c8c24df228ee4869535dcbfa288">MatrixXcd</a> A = X + X.adjoint();</div>
<div class="line">cout << <span class="stringliteral">"Here is a random self-adjoint 4x4 matrix:"</span> << endl << A << endl << endl;</div>
<div class="line"></div>
<div class="line">Tridiagonalization<MatrixXcd> triOfA(A);</div>
<div class="line">MatrixXd T = triOfA.matrixT();</div>
<div class="line">cout << <span class="stringliteral">"The tridiagonal matrix T is:"</span> << endl << T << endl << endl;</div>
<div class="line"></div>
<div class="line">cout << <span class="stringliteral">"We can also extract the diagonals of T directly ..."</span> << endl;</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga3da45e59796fbacf67fa568297927bd1">VectorXd</a> diag = triOfA.diagonal();</div>
<div class="line">cout << <span class="stringliteral">"The diagonal is:"</span> << endl << diag << endl; </div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga3da45e59796fbacf67fa568297927bd1">VectorXd</a> subdiag = triOfA.subDiagonal();</div>
<div class="line">cout << <span class="stringliteral">"The subdiagonal is:"</span> << endl << subdiag << endl;</div>
</div><!-- fragment --><p> Output: </p>
<pre class="fragment">Here is a random self-adjoint 4x4 matrix:
(-0.422,0) (0.705,-1.01) (-0.17,-0.552) (0.338,-0.357)
(0.705,1.01) (0.515,0) (0.241,-0.446) (0.05,-1.64)
(-0.17,0.552) (0.241,0.446) (-1.03,0) (0.0449,1.72)
(0.338,0.357) (0.05,1.64) (0.0449,-1.72) (1.36,0)
The tridiagonal matrix T is:
-0.422 -1.45 0 0
-1.45 1.01 -1.42 0
0 -1.42 1.8 -1.2
0 0 -1.2 -1.96
We can also extract the diagonals of T directly ...
The diagonal is:
-0.422
1.01
1.8
-1.96
The subdiagonal is:
-1.45
-1.42
-1.2
</pre><dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1Tridiagonalization.html#aceb0f16a166f4c236a1b536b7424d292" title="Returns an expression of the tridiagonal matrix T in the decomposition. ">matrixT()</a>, <a class="el" href="classEigen_1_1Tridiagonalization.html#a8fa49216273ab7579b7bea06debb1e51" title="Returns the subdiagonal of the tridiagonal matrix T in the decomposition. ">subDiagonal()</a> </dd></dl>
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<td class="memname"><a class="el" href="classEigen_1_1Matrix.html">CoeffVectorType</a> householderCoefficients </td>
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<p>Returns the Householder coefficients. </p>
<dl class="section return"><dt>Returns</dt><dd>a const reference to the vector of Householder coefficients</dd></dl>
<dl class="section pre"><dt>Precondition</dt><dd>Either the constructor <a class="el" href="classEigen_1_1Tridiagonalization.html#aa9f9722d2cef9425e2c0da3553dfbac7" title="Constructor; computes tridiagonal decomposition of given matrix. ">Tridiagonalization(const MatrixType&)</a> or the member function <a class="el" href="classEigen_1_1Tridiagonalization.html#aa69e607a4aab4fb6321ca6acbf074fc2" title="Computes tridiagonal decomposition of given matrix. ">compute(const MatrixType&)</a> has been called before to compute the tridiagonal decomposition of a matrix.</dd></dl>
<p>The Householder coefficients allow the reconstruction of the matrix <img class="formulaInl" alt="$ Q $" src="form_73.png"/> in the tridiagonal decomposition from the packed data.</p>
<p>Example: </p>
<div class="fragment"><div class="line"><a class="code" href="group__matrixtypedefs.html#gacd860ff07358f6a703c2c0d4a174e920">Matrix4d</a> X = <a class="code" href="classEigen_1_1DenseBase.html#a8e759dafdd9ecc446d397b7f5435f60a">Matrix4d::Random</a>(4,4);</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#gacd860ff07358f6a703c2c0d4a174e920">Matrix4d</a> A = X + X.transpose();</div>
<div class="line">cout << <span class="stringliteral">"Here is a random symmetric 4x4 matrix:"</span> << endl << A << endl;</div>
<div class="line">Tridiagonalization<Matrix4d> triOfA(A);</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga2006332f6989f501762673e21f5128f5">Vector3d</a> hc = triOfA.householderCoefficients();</div>
<div class="line">cout << <span class="stringliteral">"The vector of Householder coefficients is:"</span> << endl << hc << endl;</div>
</div><!-- fragment --><p> Output: </p>
<pre class="fragment">Here is a random symmetric 4x4 matrix:
1.36 0.612 0.122 0.326
0.612 -1.21 -0.222 0.563
0.122 -0.222 -0.0904 1.16
0.326 0.563 1.16 1.66
The vector of Householder coefficients is:
1.87
1.24
0
</pre><dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1Tridiagonalization.html#a66adece364b64b26b3771662de70f2df" title="Returns the internal representation of the decomposition. ">packedMatrix()</a>, <a class="el" href="group__Householder__Module.html">Householder module</a> </dd></dl>
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<td class="memname"><a class="el" href="classEigen_1_1Tridiagonalization.html#aa96bdbc1b19c647e3372c31301ea4999">HouseholderSequenceType</a> matrixQ </td>
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<p>Returns the unitary matrix Q in the decomposition. </p>
<dl class="section return"><dt>Returns</dt><dd>object representing the matrix Q</dd></dl>
<dl class="section pre"><dt>Precondition</dt><dd>Either the constructor <a class="el" href="classEigen_1_1Tridiagonalization.html#aa9f9722d2cef9425e2c0da3553dfbac7" title="Constructor; computes tridiagonal decomposition of given matrix. ">Tridiagonalization(const MatrixType&)</a> or the member function <a class="el" href="classEigen_1_1Tridiagonalization.html#aa69e607a4aab4fb6321ca6acbf074fc2" title="Computes tridiagonal decomposition of given matrix. ">compute(const MatrixType&)</a> has been called before to compute the tridiagonal decomposition of a matrix.</dd></dl>
<p>This function returns a light-weight object of template class <a class="el" href="classEigen_1_1HouseholderSequence.html" title="Sequence of Householder reflections acting on subspaces with decreasing size. ">HouseholderSequence</a>. You can either apply it directly to a matrix or you can convert it to a matrix of type <a class="el" href="classEigen_1_1Tridiagonalization.html#aeb6c0eb89cc982629305f6c7e0791caf" title="Synonym for the template parameter _MatrixType. ">MatrixType</a>.</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1Tridiagonalization.html#aa9f9722d2cef9425e2c0da3553dfbac7" title="Constructor; computes tridiagonal decomposition of given matrix. ">Tridiagonalization(const MatrixType&)</a> for an example, <a class="el" href="classEigen_1_1Tridiagonalization.html#aceb0f16a166f4c236a1b536b7424d292" title="Returns an expression of the tridiagonal matrix T in the decomposition. ">matrixT()</a>, class <a class="el" href="classEigen_1_1HouseholderSequence.html" title="Sequence of Householder reflections acting on subspaces with decreasing size. ">HouseholderSequence</a> </dd></dl>
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<td class="memname">MatrixTReturnType matrixT </td>
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<p>Returns an expression of the tridiagonal matrix T in the decomposition. </p>
<dl class="section return"><dt>Returns</dt><dd>expression object representing the matrix T</dd></dl>
<dl class="section pre"><dt>Precondition</dt><dd>Either the constructor <a class="el" href="classEigen_1_1Tridiagonalization.html#aa9f9722d2cef9425e2c0da3553dfbac7" title="Constructor; computes tridiagonal decomposition of given matrix. ">Tridiagonalization(const MatrixType&)</a> or the member function <a class="el" href="classEigen_1_1Tridiagonalization.html#aa69e607a4aab4fb6321ca6acbf074fc2" title="Computes tridiagonal decomposition of given matrix. ">compute(const MatrixType&)</a> has been called before to compute the tridiagonal decomposition of a matrix.</dd></dl>
<p>Currently, this function can be used to extract the matrix T from internal data and copy it to a dense matrix object. In most cases, it may be sufficient to directly use the packed matrix or the vector expressions returned by <a class="el" href="classEigen_1_1Tridiagonalization.html#ac109eefddd733d8e82841da5bb2dd8d3" title="Returns the diagonal of the tridiagonal matrix T in the decomposition. ">diagonal()</a> and <a class="el" href="classEigen_1_1Tridiagonalization.html#a8fa49216273ab7579b7bea06debb1e51" title="Returns the subdiagonal of the tridiagonal matrix T in the decomposition. ">subDiagonal()</a> instead of creating a new dense copy matrix with this function.</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1Tridiagonalization.html#aa9f9722d2cef9425e2c0da3553dfbac7" title="Constructor; computes tridiagonal decomposition of given matrix. ">Tridiagonalization(const MatrixType&)</a> for an example, <a class="el" href="classEigen_1_1Tridiagonalization.html#ad13845d7490115664924b3dc208ec369" title="Returns the unitary matrix Q in the decomposition. ">matrixQ()</a>, <a class="el" href="classEigen_1_1Tridiagonalization.html#a66adece364b64b26b3771662de70f2df" title="Returns the internal representation of the decomposition. ">packedMatrix()</a>, <a class="el" href="classEigen_1_1Tridiagonalization.html#ac109eefddd733d8e82841da5bb2dd8d3" title="Returns the diagonal of the tridiagonal matrix T in the decomposition. ">diagonal()</a>, <a class="el" href="classEigen_1_1Tridiagonalization.html#a8fa49216273ab7579b7bea06debb1e51" title="Returns the subdiagonal of the tridiagonal matrix T in the decomposition. ">subDiagonal()</a> </dd></dl>
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<td class="memname">const <a class="el" href="classEigen_1_1Tridiagonalization.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a>& packedMatrix </td>
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<p>Returns the internal representation of the decomposition. </p>
<dl class="section return"><dt>Returns</dt><dd>a const reference to a matrix with the internal representation of the decomposition.</dd></dl>
<dl class="section pre"><dt>Precondition</dt><dd>Either the constructor <a class="el" href="classEigen_1_1Tridiagonalization.html#aa9f9722d2cef9425e2c0da3553dfbac7" title="Constructor; computes tridiagonal decomposition of given matrix. ">Tridiagonalization(const MatrixType&)</a> or the member function <a class="el" href="classEigen_1_1Tridiagonalization.html#aa69e607a4aab4fb6321ca6acbf074fc2" title="Computes tridiagonal decomposition of given matrix. ">compute(const MatrixType&)</a> has been called before to compute the tridiagonal decomposition of a matrix.</dd></dl>
<p>The returned matrix contains the following information:</p>
<ul>
<li>the strict upper triangular part is equal to the input matrix A.</li>
<li>the diagonal and lower sub-diagonal represent the real tridiagonal symmetric matrix T.</li>
<li>the rest of the lower part contains the Householder vectors that, combined with Householder coefficients returned by <a class="el" href="classEigen_1_1Tridiagonalization.html#a2ab889c75460c178d941ee24e371b206" title="Returns the Householder coefficients. ">householderCoefficients()</a>, allows to reconstruct the matrix Q as <img class="formulaInl" alt="$ Q = H_{N-1} \ldots H_1 H_0 $" src="form_80.png"/>. Here, the matrices <img class="formulaInl" alt="$ H_i $" src="form_81.png"/> are the Householder transformations <img class="formulaInl" alt="$ H_i = (I - h_i v_i v_i^T) $" src="form_82.png"/> where <img class="formulaInl" alt="$ h_i $" src="form_83.png"/> is the <img class="formulaInl" alt="$ i $" src="form_84.png"/>th Householder coefficient and <img class="formulaInl" alt="$ v_i $" src="form_85.png"/> is the Householder vector defined by <img class="formulaInl" alt="$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T $" src="form_86.png"/> with M the matrix returned by this function.</li>
</ul>
<p>See LAPACK for further details on this packed storage.</p>
<p>Example: </p>
<div class="fragment"><div class="line"><a class="code" href="group__matrixtypedefs.html#gacd860ff07358f6a703c2c0d4a174e920">Matrix4d</a> X = <a class="code" href="classEigen_1_1DenseBase.html#a8e759dafdd9ecc446d397b7f5435f60a">Matrix4d::Random</a>(4,4);</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#gacd860ff07358f6a703c2c0d4a174e920">Matrix4d</a> A = X + X.transpose();</div>
<div class="line">cout << <span class="stringliteral">"Here is a random symmetric 4x4 matrix:"</span> << endl << A << endl;</div>
<div class="line">Tridiagonalization<Matrix4d> triOfA(A);</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#gacd860ff07358f6a703c2c0d4a174e920">Matrix4d</a> pm = triOfA.packedMatrix();</div>
<div class="line">cout << <span class="stringliteral">"The packed matrix M is:"</span> << endl << pm << endl;</div>
<div class="line">cout << <span class="stringliteral">"The diagonal and subdiagonal corresponds to the matrix T, which is:"</span> </div>
<div class="line"> << endl << triOfA.matrixT() << endl;</div>
</div><!-- fragment --><p> Output: </p>
<pre class="fragment">Here is a random symmetric 4x4 matrix:
1.36 0.612 0.122 0.326
0.612 -1.21 -0.222 0.563
0.122 -0.222 -0.0904 1.16
0.326 0.563 1.16 1.66
The packed matrix M is:
1.36 0.612 0.122 0.326
-0.704 0.0147 -0.222 0.563
0.0925 1.71 0.856 1.16
0.248 0.785 0.641 -0.506
The diagonal and subdiagonal corresponds to the matrix T, which is:
1.36 -0.704 0 0
-0.704 0.0147 1.71 0
0 1.71 0.856 0.641
0 0 0.641 -0.506
</pre><dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1Tridiagonalization.html#a2ab889c75460c178d941ee24e371b206" title="Returns the Householder coefficients. ">householderCoefficients()</a> </dd></dl>
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<td class="memname"><a class="el" href="classEigen_1_1Tridiagonalization.html">Tridiagonalization</a>< <a class="el" href="classEigen_1_1Tridiagonalization.html#aeb6c0eb89cc982629305f6c7e0791caf">MatrixType</a> >::SubDiagonalReturnType subDiagonal </td>
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<p>Returns the subdiagonal of the tridiagonal matrix T in the decomposition. </p>
<dl class="section return"><dt>Returns</dt><dd>expression representing the subdiagonal of T</dd></dl>
<dl class="section pre"><dt>Precondition</dt><dd>Either the constructor <a class="el" href="classEigen_1_1Tridiagonalization.html#aa9f9722d2cef9425e2c0da3553dfbac7" title="Constructor; computes tridiagonal decomposition of given matrix. ">Tridiagonalization(const MatrixType&)</a> or the member function <a class="el" href="classEigen_1_1Tridiagonalization.html#aa69e607a4aab4fb6321ca6acbf074fc2" title="Computes tridiagonal decomposition of given matrix. ">compute(const MatrixType&)</a> has been called before to compute the tridiagonal decomposition of a matrix.</dd></dl>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1Tridiagonalization.html#ac109eefddd733d8e82841da5bb2dd8d3" title="Returns the diagonal of the tridiagonal matrix T in the decomposition. ">diagonal()</a> for an example, <a class="el" href="classEigen_1_1Tridiagonalization.html#aceb0f16a166f4c236a1b536b7424d292" title="Returns an expression of the tridiagonal matrix T in the decomposition. ">matrixT()</a> </dd></dl>
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<hr/>The documentation for this class was generated from the following file:<ul>
<li><a class="el" href="Tridiagonalization_8h_source.html">Tridiagonalization.h</a></li>
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