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<div class="title">HouseholderQR< MatrixType > Class Template Reference<div class="ingroups"><a class="el" href="group__QR__Module.html">QR 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::HouseholderQR< MatrixType ></h3>
<p>Householder QR decomposition of a matrix. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">MatrixType</td><td>the type of the matrix of which we are computing the QR decomposition</td></tr>
</table>
</dd>
</dl>
<p>This class performs a QR decomposition of a matrix <b>A</b> into matrices <b>Q</b> and <b>R</b> such that </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \mathbf{A} = \mathbf{Q} \, \mathbf{R} \]" src="form_158.png"/>
</p>
<p> by using Householder transformations. Here, <b>Q</b> a unitary matrix and <b>R</b> an upper triangular matrix. The result is stored in a compact way compatible with LAPACK.</p>
<p>Note that no pivoting is performed. This is <b>not</b> a rank-revealing decomposition. If you want that feature, use <a class="el" href="classEigen_1_1FullPivHouseholderQR.html" title="Householder rank-revealing QR decomposition of a matrix with full pivoting. ">FullPivHouseholderQR</a> or <a class="el" href="classEigen_1_1ColPivHouseholderQR.html" title="Householder rank-revealing QR decomposition of a matrix with column-pivoting. ">ColPivHouseholderQR</a> instead.</p>
<p>This Householder QR decomposition is faster, but less numerically stable and less feature-full than <a class="el" href="classEigen_1_1FullPivHouseholderQR.html" title="Householder rank-revealing QR decomposition of a matrix with full pivoting. ">FullPivHouseholderQR</a> or <a class="el" href="classEigen_1_1ColPivHouseholderQR.html" title="Householder rank-revealing QR decomposition of a matrix with column-pivoting. ">ColPivHouseholderQR</a>.</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1MatrixBase.html#abafaafa2e7eb9f485b282041833034bf">MatrixBase::householderQr()</a> </dd></dl>
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Public Member Functions</h2></td></tr>
<tr class="memitem:a2bb8fa5593d2f447c16ba2e3cfb9c432"><td class="memItemLeft" align="right" valign="top">MatrixType::RealScalar </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderQR.html#a2bb8fa5593d2f447c16ba2e3cfb9c432">absDeterminant</a> () const </td></tr>
<tr class="separator:a2bb8fa5593d2f447c16ba2e3cfb9c432"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a5851b8e7d2a7f91592e7fa69f8b7684c"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1HouseholderQR.html">HouseholderQR</a> & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderQR.html#a5851b8e7d2a7f91592e7fa69f8b7684c">compute</a> (const MatrixType &matrix)</td></tr>
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<tr class="memitem:a321e96844544eacde2272c3a487048e9"><td class="memItemLeft" align="right" valign="top">const HCoeffsType & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderQR.html#a321e96844544eacde2272c3a487048e9">hCoeffs</a> () const </td></tr>
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<tr class="memitem:a520fc76215d7722387a76e7b7a484ee0"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequenceType</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderQR.html#a520fc76215d7722387a76e7b7a484ee0">householderQ</a> () const </td></tr>
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<tr class="memitem:ab7e9b80266fe7d1592e4c9de652d9f50"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderQR.html#ab7e9b80266fe7d1592e4c9de652d9f50">HouseholderQR</a> ()</td></tr>
<tr class="memdesc:ab7e9b80266fe7d1592e4c9de652d9f50"><td class="mdescLeft"> </td><td class="mdescRight">Default Constructor. <a href="#ab7e9b80266fe7d1592e4c9de652d9f50">More...</a><br/></td></tr>
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<tr class="memitem:abf25ea0833fe1e846c20968103f69141"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderQR.html#abf25ea0833fe1e846c20968103f69141">HouseholderQR</a> (Index rows, Index cols)</td></tr>
<tr class="memdesc:abf25ea0833fe1e846c20968103f69141"><td class="mdescLeft"> </td><td class="mdescRight">Default Constructor with memory preallocation. <a href="#abf25ea0833fe1e846c20968103f69141">More...</a><br/></td></tr>
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<tr class="memitem:ab592f6a05d7ba94c0108382baa8c993e"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderQR.html#ab592f6a05d7ba94c0108382baa8c993e">HouseholderQR</a> (const MatrixType &matrix)</td></tr>
<tr class="memdesc:ab592f6a05d7ba94c0108382baa8c993e"><td class="mdescLeft"> </td><td class="mdescRight">Constructs a QR factorization from a given matrix. <a href="#ab592f6a05d7ba94c0108382baa8c993e">More...</a><br/></td></tr>
<tr class="separator:ab592f6a05d7ba94c0108382baa8c993e"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a82a3f15f0cda1f4f874d50951344b5a9"><td class="memItemLeft" align="right" valign="top">MatrixType::RealScalar </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderQR.html#a82a3f15f0cda1f4f874d50951344b5a9">logAbsDeterminant</a> () const </td></tr>
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<tr class="memitem:a7b142db8924dd7fad99acbdd0ff4bdd1"><td class="memItemLeft" align="right" valign="top">const MatrixType & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderQR.html#a7b142db8924dd7fad99acbdd0ff4bdd1">matrixQR</a> () const </td></tr>
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<tr class="memitem:a3912046746c14d7b4e9dba999da46a68"><td class="memTemplParams" colspan="2">template<typename Rhs > </td></tr>
<tr class="memitem:a3912046746c14d7b4e9dba999da46a68"><td class="memTemplItemLeft" align="right" valign="top">const internal::solve_retval<br class="typebreak"/>
< <a class="el" href="classEigen_1_1HouseholderQR.html">HouseholderQR</a>, Rhs > </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderQR.html#a3912046746c14d7b4e9dba999da46a68">solve</a> (const <a class="el" href="classEigen_1_1MatrixBase.html">MatrixBase</a>< Rhs > &b) const </td></tr>
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<h2 class="groupheader">Constructor & Destructor Documentation</h2>
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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_1HouseholderQR.html#a5851b8e7d2a7f91592e7fa69f8b7684c">HouseholderQR::compute(const MatrixType&)</a>. </p>
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<td class="paramname"><em>rows</em>, </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_1HouseholderQR.html#ab7e9b80266fe7d1592e4c9de652d9f50" title="Default Constructor. ">HouseholderQR()</a> </dd></dl>
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<td class="paramname"><em>matrix</em></td><td>)</td>
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<p>Constructs a QR factorization from a given matrix. </p>
<p>This constructor computes the QR factorization of the matrix <em>matrix</em> by calling the method <a class="el" href="classEigen_1_1HouseholderQR.html#a5851b8e7d2a7f91592e7fa69f8b7684c">compute()</a>. It is a short cut for:</p>
<div class="fragment"><div class="line">* HouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());</div>
<div class="line">* qr.compute(matrix);</div>
<div class="line">* </div>
</div><!-- fragment --><dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1HouseholderQR.html#a5851b8e7d2a7f91592e7fa69f8b7684c">compute()</a> </dd></dl>
<p>References <a class="el" href="classEigen_1_1HouseholderQR.html#a5851b8e7d2a7f91592e7fa69f8b7684c">HouseholderQR< MatrixType >::compute()</a>.</p>
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<h2 class="groupheader">Member Function Documentation</h2>
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<td class="memname">MatrixType::RealScalar absDeterminant </td>
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<dl class="section return"><dt>Returns</dt><dd>the absolute value of the determinant of the matrix of which *this is the QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR decomposition has already been computed.</dd></dl>
<dl class="section note"><dt>Note</dt><dd>This is only for square matrices.</dd></dl>
<dl class="section warning"><dt>Warning</dt><dd>a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow. One way to work around that is to use <a class="el" href="classEigen_1_1HouseholderQR.html#a82a3f15f0cda1f4f874d50951344b5a9">logAbsDeterminant()</a> instead.</dd></dl>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1HouseholderQR.html#a82a3f15f0cda1f4f874d50951344b5a9">logAbsDeterminant()</a>, <a class="el" href="classEigen_1_1MatrixBase.html#ad63cea11a4bf220298dce4489a1704c7">MatrixBase::determinant()</a> </dd></dl>
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<td class="memname"><a class="el" href="classEigen_1_1HouseholderQR.html">HouseholderQR</a>< MatrixType > & compute </td>
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<p>Performs the QR factorization of the given matrix <em>matrix</em>. The result of the factorization is stored into <code>*this</code>, and a reference to <code>*this</code> is returned.</p>
<dl class="section see"><dt>See Also</dt><dd>class <a class="el" href="classEigen_1_1HouseholderQR.html" title="Householder QR decomposition of a matrix. ">HouseholderQR</a>, <a class="el" href="classEigen_1_1HouseholderQR.html#ab592f6a05d7ba94c0108382baa8c993e" title="Constructs a QR factorization from a given matrix. ">HouseholderQR(const MatrixType&)</a> </dd></dl>
<p>Referenced by <a class="el" href="classEigen_1_1HouseholderQR.html#ab592f6a05d7ba94c0108382baa8c993e">HouseholderQR< MatrixType >::HouseholderQR()</a>.</p>
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<td class="memname">const HCoeffsType& hCoeffs </td>
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<dl class="section return"><dt>Returns</dt><dd>a const reference to the vector of Householder coefficients used to represent the factor <code>Q</code>.</dd></dl>
<p>For advanced uses only. </p>
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<td class="memname"><a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequenceType</a> householderQ </td>
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<p>This method returns an expression of the unitary matrix Q as a sequence of Householder transformations.</p>
<p>The returned expression can directly be used to perform matrix products. It can also be assigned to a dense <a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> object. Here is an example showing how to recover the full or thin matrix Q, as well as how to perform matrix products using operator*:</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>(5,3)), thinQ(<a class="code" href="classEigen_1_1MatrixBase.html#a0650b65c6ae6c3d19a138b72a6d68568">MatrixXf::Identity</a>(5,3)), Q;</div>
<div class="line">A.setRandom();</div>
<div class="line">HouseholderQR<MatrixXf> qr(A);</div>
<div class="line">Q = qr.householderQ();</div>
<div class="line">thinQ = qr.householderQ() * thinQ;</div>
<div class="line">std::cout << <span class="stringliteral">"The complete unitary matrix Q is:\n"</span> << Q << <span class="stringliteral">"\n\n"</span>;</div>
<div class="line">std::cout << <span class="stringliteral">"The thin matrix Q is:\n"</span> << thinQ << <span class="stringliteral">"\n\n"</span>;</div>
</div><!-- fragment --><p> Output: </p>
<pre class="fragment">The complete unitary matrix Q is:
-0.676 0.0793 0.713 -0.0788 -0.147
-0.221 -0.322 -0.37 -0.366 -0.759
-0.353 -0.345 -0.214 0.841 -0.0518
0.582 -0.462 0.555 0.176 -0.329
-0.174 -0.747 -0.00907 -0.348 0.539
The thin matrix Q is:
-0.676 0.0793 0.713
-0.221 -0.322 -0.37
-0.353 -0.345 -0.214
0.582 -0.462 0.555
-0.174 -0.747 -0.00907
</pre>
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<td class="memname">MatrixType::RealScalar logAbsDeterminant </td>
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<dl class="section return"><dt>Returns</dt><dd>the natural log of the absolute value of the determinant of the matrix of which *this is the QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR decomposition has already been computed.</dd></dl>
<dl class="section note"><dt>Note</dt><dd>This is only for square matrices.</dd>
<dd>
This method is useful to work around the risk of overflow/underflow that's inherent to determinant computation.</dd></dl>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1HouseholderQR.html#a2bb8fa5593d2f447c16ba2e3cfb9c432">absDeterminant()</a>, <a class="el" href="classEigen_1_1MatrixBase.html#ad63cea11a4bf220298dce4489a1704c7">MatrixBase::determinant()</a> </dd></dl>
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<td class="memname">const MatrixType& matrixQR </td>
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<td> const</td>
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<dl class="section return"><dt>Returns</dt><dd>a reference to the matrix where the Householder QR decomposition is stored in a LAPACK-compatible way. </dd></dl>
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<td class="memname">const internal::solve_retval<<a class="el" href="classEigen_1_1HouseholderQR.html">HouseholderQR</a>, Rhs> solve </td>
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<td class="paramtype">const <a class="el" href="classEigen_1_1MatrixBase.html">MatrixBase</a>< Rhs > & </td>
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<p>This method finds a solution x to the equation Ax=b, where A is the matrix of which *this is the QR decomposition, if any exists.</p>
<dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramname">b</td><td>the right-hand-side of the equation to solve.</td></tr>
</table>
</dd>
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<dl class="section return"><dt>Returns</dt><dd>a solution.</dd></dl>
<dl class="section note"><dt>Note</dt><dd>The case where b is a matrix is not yet implemented. Also, this code is space inefficient.</dd></dl>
<p>This method just tries to find as good a solution as possible. If you want to check whether a solution exists or if it is accurate, just call this function to get a result and then compute the error of this result, or use <a class="el" href="classEigen_1_1DenseBase.html#a158c2184951e6e415c2e9b98db8e8966">MatrixBase::isApprox()</a> directly, for instance like this:</p>
<div class="fragment"><div class="line"><span class="keywordtype">bool</span> a_solution_exists = (A*result).isApprox(b, precision); </div>
</div><!-- fragment --><p> This method avoids dividing by zero, so that the non-existence of a solution doesn't by itself mean that you'll get <code>inf</code> or <code>nan</code> values.</p>
<p>If there exists more than one solution, this method will arbitrarily choose one.</p>
<p>Example: </p>
<div class="fragment"><div class="line"><span class="keyword">typedef</span> Matrix<float,3,3> Matrix3x3;</div>
<div class="line">Matrix3x3 m = Matrix3x3::Random();</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga2b787393a7fc870d99aa634f60b2510c">Matrix3f</a> y = <a class="code" href="classEigen_1_1DenseBase.html#a8e759dafdd9ecc446d397b7f5435f60a">Matrix3f::Random</a>();</div>
<div class="line">cout << <span class="stringliteral">"Here is the matrix m:"</span> << endl << m << endl;</div>
<div class="line">cout << <span class="stringliteral">"Here is the matrix y:"</span> << endl << y << endl;</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga2b787393a7fc870d99aa634f60b2510c">Matrix3f</a> x;</div>
<div class="line">x = m.householderQr().solve(y);</div>
<div class="line">assert(y.isApprox(m*x));</div>
<div class="line">cout << <span class="stringliteral">"Here is a solution x to the equation mx=y:"</span> << endl << x << endl;</div>
</div><!-- fragment --><p> Output: </p>
<pre class="fragment">Here is the matrix m:
0.68 0.597 -0.33
-0.211 0.823 0.536
0.566 -0.605 -0.444
Here is the matrix y:
0.108 -0.27 0.832
-0.0452 0.0268 0.271
0.258 0.904 0.435
Here is a solution x to the equation mx=y:
0.609 2.68 1.67
-0.231 -1.57 0.0713
0.51 3.51 1.05
</pre>
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<hr/>The documentation for this class was generated from the following files:<ul>
<li><a class="el" href="ForwardDeclarations_8h_source.html">ForwardDeclarations.h</a></li>
<li><a class="el" href="HouseholderQR_8h_source.html">HouseholderQR.h</a></li>
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