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<div id="nav-toc" class="toc"><h3>Table of contents</h3><ul><li class="level1"><a id="link0" href="#title0" title="H2">Detailed Description</a></li><li class="level1"><a id="link1" href="#title1" title="H2">
Public Member Functions</a></li><li class="level1"><a id="link2" href="#title2" title="H2">
Additional Inherited Members</a></li><li class="level1"><a id="link3" href="#title3" title="H2">Constructor & Destructor Documentation</a></li><li class="level1"><a id="link4" href="#title4" title="H2">◆ FullPivLU() [1/4]</a></li><li class="level1"><a id="link5" href="#title5" title="H2">◆ FullPivLU() [2/4]</a></li><li class="level1"><a id="link6" href="#title6" title="H2">◆ FullPivLU() [3/4]</a></li><li class="level1"><a id="link7" href="#title7" title="H2">◆ FullPivLU() [4/4]</a></li><li class="level1"><a id="link8" href="#title8" title="H2">Member Function Documentation</a></li><li class="level1"><a id="link9" href="#title9" title="H2">◆ compute()</a></li><li class="level1"><a id="link10" href="#title10" title="H2">◆ determinant()</a></li><li class="level1"><a id="link11" href="#title11" title="H2">◆ dimensionOfKernel()</a></li><li class="level1"><a id="link12" href="#title12" title="H2">◆ image()</a></li><li class="level1"><a id="link13" href="#title13" title="H2">◆ inverse()</a></li><li class="level1"><a id="link14" href="#title14" title="H2">◆ isInjective()</a></li><li class="level1"><a id="link15" href="#title15" title="H2">◆ isInvertible()</a></li><li class="level1"><a id="link16" href="#title16" title="H2">◆ isSurjective()</a></li><li class="level1"><a id="link17" href="#title17" title="H2">◆ kernel()</a></li><li class="level1"><a id="link18" href="#title18" title="H2">◆ matrixLU()</a></li><li class="level1"><a id="link19" href="#title19" title="H2">◆ maxPivot()</a></li><li class="level1"><a id="link20" href="#title20" title="H2">◆ nonzeroPivots()</a></li><li class="level1"><a id="link21" href="#title21" title="H2">◆ permutationP()</a></li><li class="level1"><a id="link22" href="#title22" title="H2">◆ permutationQ()</a></li><li class="level1"><a id="link23" href="#title23" title="H2">◆ rank()</a></li><li class="level1"><a id="link24" href="#title24" title="H2">◆ rcond()</a></li><li class="level1"><a id="link25" href="#title25" title="H2">◆ reconstructedMatrix()</a></li><li class="level1"><a id="link26" href="#title26" title="H2">◆ setThreshold() [1/2]</a></li><li class="level1"><a id="link27" href="#title27" title="H2">◆ setThreshold() [2/2]</a></li><li class="level1"><a id="link28" href="#title28" title="H2">◆ solve()</a></li><li class="level1"><a id="link29" href="#title29" title="H2">◆ threshold()</a></li></ul></div><div class="ui-resizable-handle ui-resizable-e" style="z-index: 90;"></div><div class="ui-resizable-handle ui-resizable-s" style="z-index: 90;"></div><div class="ui-resizable-handle ui-resizable-se ui-icon ui-icon-gripsmall-diagonal-se" style="z-index: 90;"></div></div>
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<div class="title">Eigen::FullPivLU< MatrixType_ > Class Template Reference<div class="ingroups"><a class="el" href="group__DenseLinearSolvers__chapter.html">Dense linear problems and decompositions</a> » <a class="el" href="group__DenseLinearSolvers__Reference.html">Reference</a> » <a class="el" href="group__LU__Module.html">LU module</a></div></div> </div>
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<a name="details" id="details"></a><h2 class="groupheader anchor" id="title0">Detailed Description</h2>
<div class="textblock"><h3>template<typename MatrixType_><br>
class Eigen::FullPivLU< MatrixType_ ></h3>
<p>LU decomposition of a matrix with complete pivoting, and related features. </p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tbody><tr><td class="paramname">MatrixType_</td><td>the type of the matrix of which we are computing the LU decomposition</td></tr>
</tbody></table>
</dd>
</dl>
<p>This class represents a LU decomposition of any matrix, with complete pivoting: the matrix A is decomposed as <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0" style=""><nobr><span class="math" id="MathJax-Span-1" style="width: 8.727em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.146em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.174em, 1007.15em, 2.52em, -999.997em); top: -2.163em; left: 0em;"><span class="mrow" id="MathJax-Span-2"><span class="mi" id="MathJax-Span-3" style="font-family: MathJax_Math-italic;">A</span><span class="mo" id="MathJax-Span-4" style="font-family: MathJax_Main; padding-left: 0.296em;">=</span><span class="msubsup" id="MathJax-Span-5" style="padding-left: 0.296em;"><span style="display: inline-block; position: relative; width: 1.818em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.76em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mi" id="MathJax-Span-6" style="font-family: MathJax_Math-italic;">P<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.33em; left: 0.823em;"><span class="texatom" id="MathJax-Span-7"><span class="mrow" id="MathJax-Span-8"><span class="mo" id="MathJax-Span-9" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-10" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mi" id="MathJax-Span-11" style="font-family: MathJax_Math-italic;">L</span><span class="mi" id="MathJax-Span-12" style="font-family: MathJax_Math-italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.061em;"></span></span><span class="msubsup" id="MathJax-Span-13"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.76em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mi" id="MathJax-Span-14" style="font-family: MathJax_Math-italic;">Q</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.33em; left: 0.823em;"><span class="texatom" id="MathJax-Span-15"><span class="mrow" id="MathJax-Span-16"><span class="mo" id="MathJax-Span-17" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-18" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.169em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.432em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-1"> A = P^{-1} L U Q^{-1} </script> where L is unit-lower-triangular, U is upper-triangular, and P and Q are permutation matrices. This is a rank-revealing LU decomposition. The eigenvalues (diagonal coefficients) of U are sorted in such a way that any zeros are at the end.</p>
<p>This decomposition provides the generic approach to solving systems of linear equations, computing the rank, invertibility, inverse, kernel, and determinant.</p>
<p>This LU decomposition is very stable and well tested with large matrices. However there are use cases where the SVD decomposition is inherently more stable and/or flexible. For example, when computing the kernel of a matrix, working with the SVD allows to select the smallest singular values of the matrix, something that the LU decomposition doesn't see.</p>
<p>The data of the LU decomposition can be directly accessed through the methods <a class="el" href="classEigen_1_1FullPivLU.html#a3e7d7a53f5b7c4ba99013fe171ac5654">matrixLU()</a>, <a class="el" href="classEigen_1_1FullPivLU.html#a3eb3aa0c37e06ffaf0c07b1eeb0995cc">permutationP()</a>, <a class="el" href="classEigen_1_1FullPivLU.html#a25b70ffbc88d804981c8874da55e7419">permutationQ()</a>.</p>
<p>As an example, here is how the original matrix can be retrieved: </p><div class="fragment"><div class="line"><span class="keyword">typedef</span> Matrix<double, 5, 3> Matrix5x3;</div>
<div class="line"><span class="keyword">typedef</span> Matrix<double, 5, 5> Matrix5x5;</div>
<div class="line">Matrix5x3 m = Matrix5x3::Random();</div>
<div class="line">cout << <span class="stringliteral">"Here is the matrix m:"</span> << endl << m << endl;</div>
<div class="line"><a class="code" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU<Matrix5x3></a> lu(m);</div>
<div class="line">cout << <span class="stringliteral">"Here is, up to permutations, its LU decomposition matrix:"</span></div>
<div class="line"> << endl << lu.matrixLU() << endl;</div>
<div class="line">cout << <span class="stringliteral">"Here is the L part:"</span> << endl;</div>
<div class="line">Matrix5x5 l = Matrix5x5::Identity();</div>
<div class="line">l.block<5,3>(0,0).triangularView<StrictlyLower>() = lu.matrixLU();</div>
<div class="line">cout << l << endl;</div>
<div class="line">cout << <span class="stringliteral">"Here is the U part:"</span> << endl;</div>
<div class="line">Matrix5x3 u = lu.matrixLU().triangularView<<a class="code" href="group__enums.html#gga39e3366ff5554d731e7dc8bb642f83cdafca2ccebb604f171656deb53e8c083c1">Upper</a>>();</div>
<div class="line">cout << u << endl;</div>
<div class="line">cout << <span class="stringliteral">"Let us now reconstruct the original matrix m:"</span> << endl;</div>
<div class="line">cout << lu.permutationP().inverse() * l * u * lu.permutationQ().inverse() << endl;</div>
<div class="ttc" id="aclassEigen_1_1FullPivLU_html"><div class="ttname"><a href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a></div><div class="ttdoc">LU decomposition of a matrix with complete pivoting, and related features.</div><div class="ttdef"><b>Definition:</b> FullPivLU.h:64</div></div>
<div class="ttc" id="agroup__enums_html_gga39e3366ff5554d731e7dc8bb642f83cdafca2ccebb604f171656deb53e8c083c1"><div class="ttname"><a href="group__enums.html#gga39e3366ff5554d731e7dc8bb642f83cdafca2ccebb604f171656deb53e8c083c1">Eigen::Upper</a></div><div class="ttdeci">@ Upper</div><div class="ttdef"><b>Definition:</b> Constants.h:213</div></div>
</div><!-- fragment --><p> Output: </p><pre class="fragment">Here is the matrix m:
0.68 -0.605 -0.0452
-0.211 -0.33 0.258
0.566 0.536 -0.27
0.597 -0.444 0.0268
0.823 0.108 0.904
Here is, up to permutations, its LU decomposition matrix:
0.904 0.823 0.108
-0.299 0.812 0.569
-0.05 0.888 -1.1
0.0296 0.705 0.768
0.285 -0.549 0.0436
Here is the L part:
1 0 0 0 0
-0.299 1 0 0 0
-0.05 0.888 1 0 0
0.0296 0.705 0.768 1 0
0.285 -0.549 0.0436 0 1
Here is the U part:
0.904 0.823 0.108
0 0.812 0.569
0 0 -1.1
0 0 0
0 0 0
Let us now reconstruct the original matrix m:
0.68 -0.605 -0.0452
-0.211 -0.33 0.258
0.566 0.536 -0.27
0.597 -0.444 0.0268
0.823 0.108 0.904
</pre><p>This class supports the <a class="el" href="group__InplaceDecomposition.html">inplace decomposition </a> mechanism.</p>
<dl class="section see"><dt>See also</dt><dd><a class="el" href="classEigen_1_1MatrixBase.html#a25da97d31acab0ee5d9d13bdbb0569da">MatrixBase::fullPivLu()</a>, <a class="el" href="classEigen_1_1MatrixBase.html#a7ad8f77004bb956b603bb43fd2e3c061">MatrixBase::determinant()</a>, <a class="el" href="classEigen_1_1MatrixBase.html#a7712eb69e8ea3c8f7b8da1c44dbdeebf">MatrixBase::inverse()</a> </dd></dl>
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<img id="dynsection-0-trigger" src="closed.png" alt="+"> Inheritance diagram for Eigen::FullPivLU< MatrixType_ >:</div>
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<tbody><tr class="heading"><td colspan="2"><h2 class="groupheader anchor" id="title1"><a name="pub-methods"></a>
Public Member Functions</h2></td></tr>
<tr class="memitem:a396f63d737e0613f41004e30be8fe3cf"><td class="memTemplParams" colspan="2">template<typename InputType > </td></tr>
<tr class="memitem:a396f63d737e0613f41004e30be8fe3cf"><td class="memTemplItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a> & </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a396f63d737e0613f41004e30be8fe3cf">compute</a> (const <a class="el" href="structEigen_1_1EigenBase.html">EigenBase</a>< InputType > &matrix)</td></tr>
<tr class="separator:a396f63d737e0613f41004e30be8fe3cf"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a71654e5c60a26407ecccfaa5b34bb0aa"><td class="memItemLeft" align="right" valign="top">internal::traits< MatrixType >::Scalar </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a71654e5c60a26407ecccfaa5b34bb0aa">determinant</a> () const</td></tr>
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<tr class="memitem:a358cec49914ec3cd3707e6b79ae32d0b"><td class="memItemLeft" align="right" valign="top"><a class="el" href="structEigen_1_1EigenBase.html#a554f30542cc2316add4b1ea0a492ff02">Index</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a358cec49914ec3cd3707e6b79ae32d0b">dimensionOfKernel</a> () const</td></tr>
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<tr class="memitem:af225528d1c6e623a2b1dce091907d13e"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#af225528d1c6e623a2b1dce091907d13e">FullPivLU</a> ()</td></tr>
<tr class="memdesc:af225528d1c6e623a2b1dce091907d13e"><td class="mdescLeft"> </td><td class="mdescRight">Default Constructor. <a href="classEigen_1_1FullPivLU.html#af225528d1c6e623a2b1dce091907d13e">More...</a><br></td></tr>
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<tr class="memitem:a31a6a984478a9f721f367667fe4c5ab1"><td class="memTemplParams" colspan="2">template<typename InputType > </td></tr>
<tr class="memitem:a31a6a984478a9f721f367667fe4c5ab1"><td class="memTemplItemLeft" align="right" valign="top"> </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a31a6a984478a9f721f367667fe4c5ab1">FullPivLU</a> (const <a class="el" href="structEigen_1_1EigenBase.html">EigenBase</a>< InputType > &matrix)</td></tr>
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<tr class="memitem:a3e903b9f401e3fc5d1ca7c6951c76185"><td class="memTemplParams" colspan="2">template<typename InputType > </td></tr>
<tr class="memitem:a3e903b9f401e3fc5d1ca7c6951c76185"><td class="memTemplItemLeft" align="right" valign="top"> </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a3e903b9f401e3fc5d1ca7c6951c76185">FullPivLU</a> (<a class="el" href="structEigen_1_1EigenBase.html">EigenBase</a>< InputType > &matrix)</td></tr>
<tr class="memdesc:a3e903b9f401e3fc5d1ca7c6951c76185"><td class="mdescLeft"> </td><td class="mdescRight">Constructs a LU factorization from a given matrix. <a href="classEigen_1_1FullPivLU.html#a3e903b9f401e3fc5d1ca7c6951c76185">More...</a><br></td></tr>
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<tr class="memitem:ae83ebd2a24088f04e3ac835b0dc001e1"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#ae83ebd2a24088f04e3ac835b0dc001e1">FullPivLU</a> (<a class="el" href="structEigen_1_1EigenBase.html#a554f30542cc2316add4b1ea0a492ff02">Index</a> rows, <a class="el" href="structEigen_1_1EigenBase.html#a554f30542cc2316add4b1ea0a492ff02">Index</a> cols)</td></tr>
<tr class="memdesc:ae83ebd2a24088f04e3ac835b0dc001e1"><td class="mdescLeft"> </td><td class="mdescRight">Default Constructor with memory preallocation. <a href="classEigen_1_1FullPivLU.html#ae83ebd2a24088f04e3ac835b0dc001e1">More...</a><br></td></tr>
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<tr class="memitem:aa8cbf984141608e89b503125690d24d4"><td class="memItemLeft" align="right" valign="top">const internal::image_retval< <a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a> > </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#aa8cbf984141608e89b503125690d24d4">image</a> (const MatrixType &originalMatrix) const</td></tr>
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<tr class="memitem:a34afc848d7fb22c7a56a053d3807d2cd"><td class="memItemLeft" align="right" valign="top">const <a class="el" href="classEigen_1_1Inverse.html">Inverse</a>< <a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a> > </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a34afc848d7fb22c7a56a053d3807d2cd">inverse</a> () const</td></tr>
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<tr class="memitem:a90dd33c632ba890175f61eac054bde98"><td class="memItemLeft" align="right" valign="top">bool </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a90dd33c632ba890175f61eac054bde98">isInjective</a> () const</td></tr>
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<tr class="memitem:a0ee7753645eb31bcbd5faa459168b294"><td class="memItemLeft" align="right" valign="top">bool </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a0ee7753645eb31bcbd5faa459168b294">isInvertible</a> () const</td></tr>
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<tr class="memitem:af42b9cb6356658b92b2d1006aee73fc4"><td class="memItemLeft" align="right" valign="top">bool </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#af42b9cb6356658b92b2d1006aee73fc4">isSurjective</a> () const</td></tr>
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<tr class="memitem:adfc1e27ff60287be5313b5efc3559308"><td class="memItemLeft" align="right" valign="top">const internal::kernel_retval< <a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a> > </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#adfc1e27ff60287be5313b5efc3559308">kernel</a> () const</td></tr>
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<tr class="memitem:a3e7d7a53f5b7c4ba99013fe171ac5654"><td class="memItemLeft" align="right" valign="top">const MatrixType & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a3e7d7a53f5b7c4ba99013fe171ac5654">matrixLU</a> () const</td></tr>
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<tr class="memitem:a05cedf8dca6394355ef64c1ea1374b4a"><td class="memItemLeft" align="right" valign="top">RealScalar </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a05cedf8dca6394355ef64c1ea1374b4a">maxPivot</a> () const</td></tr>
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<tr class="memitem:aad90c46ea08618ae485fce4e5f4677d0"><td class="memItemLeft" align="right" valign="top"><a class="el" href="structEigen_1_1EigenBase.html#a554f30542cc2316add4b1ea0a492ff02">Index</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#aad90c46ea08618ae485fce4e5f4677d0">nonzeroPivots</a> () const</td></tr>
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<tr class="memitem:a3eb3aa0c37e06ffaf0c07b1eeb0995cc"><td class="memItemLeft" align="right" valign="top">const <a class="el" href="classEigen_1_1PermutationMatrix.html">PermutationPType</a> & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a3eb3aa0c37e06ffaf0c07b1eeb0995cc">permutationP</a> () const</td></tr>
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<tr class="memitem:a25b70ffbc88d804981c8874da55e7419"><td class="memItemLeft" align="right" valign="top">const <a class="el" href="classEigen_1_1PermutationMatrix.html">PermutationQType</a> & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a25b70ffbc88d804981c8874da55e7419">permutationQ</a> () const</td></tr>
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<tr class="memitem:a8d31c78a17a70d56ef2d105d6b5efec3"><td class="memItemLeft" align="right" valign="top"><a class="el" href="structEigen_1_1EigenBase.html#a554f30542cc2316add4b1ea0a492ff02">Index</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a8d31c78a17a70d56ef2d105d6b5efec3">rank</a> () const</td></tr>
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<tr class="memitem:ae39fcfa8d1319472a5b2adfa7a28d9cf"><td class="memItemLeft" align="right" valign="top">RealScalar </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#ae39fcfa8d1319472a5b2adfa7a28d9cf">rcond</a> () const</td></tr>
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<tr class="memitem:a191a4f598b0c192a83ab48984e87ee51"><td class="memItemLeft" align="right" valign="top">MatrixType </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a191a4f598b0c192a83ab48984e87ee51">reconstructedMatrix</a> () const</td></tr>
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<tr class="memitem:abad257b6db0856d8ec52c6072f58f75d"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a> & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#abad257b6db0856d8ec52c6072f58f75d">setThreshold</a> (const RealScalar &<a class="el" href="classEigen_1_1FullPivLU.html#ae2298a7a89749dee7d86f02ccacce0cb">threshold</a>)</td></tr>
<tr class="separator:abad257b6db0856d8ec52c6072f58f75d"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aeafbb0b885cc4c28b53e77988ac5cfe3"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a> & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#aeafbb0b885cc4c28b53e77988ac5cfe3">setThreshold</a> (Default_t)</td></tr>
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<tr class="memitem:a614d3aa28e6af7b2af8630d3e2d022d8"><td class="memTemplParams" colspan="2">template<typename Rhs > </td></tr>
<tr class="memitem:a614d3aa28e6af7b2af8630d3e2d022d8"><td class="memTemplItemLeft" align="right" valign="top">const <a class="el" href="classEigen_1_1Solve.html">Solve</a>< <a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>, Rhs > </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a614d3aa28e6af7b2af8630d3e2d022d8">solve</a> (const <a class="el" href="classEigen_1_1MatrixBase.html">MatrixBase</a>< Rhs > &b) const</td></tr>
<tr class="separator:a614d3aa28e6af7b2af8630d3e2d022d8"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ae2298a7a89749dee7d86f02ccacce0cb"><td class="memItemLeft" align="right" valign="top">RealScalar </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#ae2298a7a89749dee7d86f02ccacce0cb">threshold</a> () const</td></tr>
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<tr class="inherit_header pub_methods_classEigen_1_1SolverBase"><td colspan="2" onclick="javascript:toggleInherit('pub_methods_classEigen_1_1SolverBase')"><img src="closed.png" alt="-"> Public Member Functions inherited from <a class="el" href="classEigen_1_1SolverBase.html">Eigen::SolverBase< FullPivLU< MatrixType_ > ></a></td></tr>
<tr class="memitem:ae1025416bdb5a768f7213c67feb4dc33 inherit pub_methods_classEigen_1_1SolverBase"><td class="memItemLeft" align="right" valign="top">const AdjointReturnType </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1SolverBase.html#ae1025416bdb5a768f7213c67feb4dc33">adjoint</a> () const</td></tr>
<tr class="separator:ae1025416bdb5a768f7213c67feb4dc33 inherit pub_methods_classEigen_1_1SolverBase"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a1fbabe7f12bcbfba3b9a448b1f5e46fa inherit pub_methods_classEigen_1_1SolverBase"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>< MatrixType_ > & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1SolverBase.html#a1fbabe7f12bcbfba3b9a448b1f5e46fa">derived</a> ()</td></tr>
<tr class="separator:a1fbabe7f12bcbfba3b9a448b1f5e46fa inherit pub_methods_classEigen_1_1SolverBase"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:afd4f3f1c57b7594b96a7e30f2974ea2e inherit pub_methods_classEigen_1_1SolverBase"><td class="memItemLeft" align="right" valign="top">const <a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>< MatrixType_ > & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1SolverBase.html#afd4f3f1c57b7594b96a7e30f2974ea2e">derived</a> () const</td></tr>
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<tr class="memitem:a7fd647d110487799205df6f99547879d inherit pub_methods_classEigen_1_1SolverBase"><td class="memItemLeft" align="right" valign="top">const <a class="el" href="classEigen_1_1Solve.html">Solve</a>< <a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>< MatrixType_ >, Rhs > </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1SolverBase.html#a7fd647d110487799205df6f99547879d">solve</a> (const <a class="el" href="classEigen_1_1MatrixBase.html">MatrixBase</a>< Rhs > &b) const</td></tr>
<tr class="separator:a7fd647d110487799205df6f99547879d inherit pub_methods_classEigen_1_1SolverBase"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a4d5e5baddfba3790ab1a5f247dcc4dc1 inherit pub_methods_classEigen_1_1SolverBase"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1SolverBase.html#a4d5e5baddfba3790ab1a5f247dcc4dc1">SolverBase</a> ()</td></tr>
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<tr class="memitem:a70cf5cd1b31dbb4f4d61c436c83df6d3 inherit pub_methods_classEigen_1_1SolverBase"><td class="memItemLeft" align="right" valign="top">const <a class="el" href="classEigen_1_1Transpose.html">ConstTransposeReturnType</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1SolverBase.html#a70cf5cd1b31dbb4f4d61c436c83df6d3">transpose</a> () const</td></tr>
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<tr class="inherit_header pub_methods_structEigen_1_1EigenBase"><td colspan="2" onclick="javascript:toggleInherit('pub_methods_structEigen_1_1EigenBase')"><img src="closed.png" alt="-"> Public Member Functions inherited from <a class="el" href="structEigen_1_1EigenBase.html">Eigen::EigenBase< Derived ></a></td></tr>
<tr class="memitem:a2d768a9877f5f69f49432d447b552bfe inherit pub_methods_structEigen_1_1EigenBase"><td class="memItemLeft" align="right" valign="top">EIGEN_CONSTEXPR <a class="el" href="structEigen_1_1EigenBase.html#a554f30542cc2316add4b1ea0a492ff02">Index</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="structEigen_1_1EigenBase.html#a2d768a9877f5f69f49432d447b552bfe">cols</a> () const EIGEN_NOEXCEPT</td></tr>
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<tr class="memitem:a1fbabe7f12bcbfba3b9a448b1f5e46fa inherit pub_methods_structEigen_1_1EigenBase"><td class="memItemLeft" align="right" valign="top">Derived & </td><td class="memItemRight" valign="bottom"><a class="el" href="structEigen_1_1EigenBase.html#a1fbabe7f12bcbfba3b9a448b1f5e46fa">derived</a> ()</td></tr>
<tr class="separator:a1fbabe7f12bcbfba3b9a448b1f5e46fa inherit pub_methods_structEigen_1_1EigenBase"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:afd4f3f1c57b7594b96a7e30f2974ea2e inherit pub_methods_structEigen_1_1EigenBase"><td class="memItemLeft" align="right" valign="top">const Derived & </td><td class="memItemRight" valign="bottom"><a class="el" href="structEigen_1_1EigenBase.html#afd4f3f1c57b7594b96a7e30f2974ea2e">derived</a> () const</td></tr>
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<tr class="memitem:ac22eb0695d00edd7d4a3b2d0a98b81c2 inherit pub_methods_structEigen_1_1EigenBase"><td class="memItemLeft" align="right" valign="top">EIGEN_CONSTEXPR <a class="el" href="structEigen_1_1EigenBase.html#a554f30542cc2316add4b1ea0a492ff02">Index</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="structEigen_1_1EigenBase.html#ac22eb0695d00edd7d4a3b2d0a98b81c2">rows</a> () const EIGEN_NOEXCEPT</td></tr>
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<tr class="memitem:ae106171b6fefd3f7af108a8283de36c9 inherit pub_methods_structEigen_1_1EigenBase"><td class="memItemLeft" align="right" valign="top">EIGEN_CONSTEXPR <a class="el" href="structEigen_1_1EigenBase.html#a554f30542cc2316add4b1ea0a492ff02">Index</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="structEigen_1_1EigenBase.html#ae106171b6fefd3f7af108a8283de36c9">size</a> () const EIGEN_NOEXCEPT</td></tr>
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<tbody><tr class="heading"><td colspan="2"><h2 class="groupheader anchor" id="title2"><a name="inherited"></a>
Additional Inherited Members</h2></td></tr>
<tr class="inherit_header pub_types_structEigen_1_1EigenBase"><td colspan="2" onclick="javascript:toggleInherit('pub_types_structEigen_1_1EigenBase')"><img src="closed.png" alt="-"> Public Types inherited from <a class="el" href="structEigen_1_1EigenBase.html">Eigen::EigenBase< Derived ></a></td></tr>
<tr class="memitem:a554f30542cc2316add4b1ea0a492ff02 inherit pub_types_structEigen_1_1EigenBase"><td class="memItemLeft" align="right" valign="top">typedef <a class="el" href="namespaceEigen.html#a62e77e0933482dafde8fe197d9a2cfde">Eigen::Index</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="structEigen_1_1EigenBase.html#a554f30542cc2316add4b1ea0a492ff02">Index</a></td></tr>
<tr class="memdesc:a554f30542cc2316add4b1ea0a492ff02 inherit pub_types_structEigen_1_1EigenBase"><td class="mdescLeft"> </td><td class="mdescRight">The interface type of indices. <a href="structEigen_1_1EigenBase.html#a554f30542cc2316add4b1ea0a492ff02">More...</a><br></td></tr>
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<h2 class="groupheader anchor" id="title3">Constructor & Destructor Documentation</h2>
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<h2 class="memtitle anchor" id="title4"><span class="permalink"><a href="#af225528d1c6e623a2b1dce091907d13e">◆ </a></span>FullPivLU() <span class="overload">[1/4]</span></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 LU::compute(const MatrixType&). </p>
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<h2 class="memtitle anchor" id="title5"><span class="permalink"><a href="#ae83ebd2a24088f04e3ac835b0dc001e1">◆ </a></span>FullPivLU() <span class="overload">[2/4]</span></h2>
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template<typename MatrixType > </div>
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<td class="memname"><a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType >::<a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a> </td>
<td>(</td>
<td class="paramtype"><a class="el" href="structEigen_1_1EigenBase.html#a554f30542cc2316add4b1ea0a492ff02">Index</a> </td>
<td class="paramname"><em>rows</em>, </td>
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<td class="paramtype"><a class="el" href="structEigen_1_1EigenBase.html#a554f30542cc2316add4b1ea0a492ff02">Index</a> </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_1FullPivLU.html#af225528d1c6e623a2b1dce091907d13e" title="Default Constructor.">FullPivLU()</a> </dd></dl>
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<h2 class="memtitle anchor" id="title6"><span class="permalink"><a href="#a31a6a984478a9f721f367667fe4c5ab1">◆ </a></span>FullPivLU() <span class="overload">[3/4]</span></h2>
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template<typename MatrixType > </div>
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<td class="memname"><a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType >::<a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a> </td>
<td>(</td>
<td class="paramtype">const <a class="el" href="structEigen_1_1EigenBase.html">EigenBase</a>< InputType > & </td>
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<p>Constructor.</p>
<dl class="params"><dt>Parameters</dt><dd>
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<tbody><tr><td class="paramname">matrix</td><td>the matrix of which to compute the LU decomposition. It is required to be nonzero. </td></tr>
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<h2 class="memtitle anchor" id="title7"><span class="permalink"><a href="#a3e903b9f401e3fc5d1ca7c6951c76185">◆ </a></span>FullPivLU() <span class="overload">[4/4]</span></h2>
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<td class="memname"><a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType >::<a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a> </td>
<td>(</td>
<td class="paramtype"><a class="el" href="structEigen_1_1EigenBase.html">EigenBase</a>< InputType > & </td>
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<p>Constructs a LU factorization from a given matrix. </p>
<p>This overloaded constructor is provided for <a class="el" href="group__InplaceDecomposition.html">inplace decomposition </a> when <code>MatrixType</code> is a <a class="el" href="classEigen_1_1Ref.html" title="A matrix or vector expression mapping an existing expression.">Eigen::Ref</a>.</p>
<dl class="section see"><dt>See also</dt><dd>FullPivLU(const EigenBase&) </dd></dl>
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<h2 class="groupheader anchor" id="title8">Member Function Documentation</h2>
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<h2 class="memtitle anchor" id="title9"><span class="permalink"><a href="#a396f63d737e0613f41004e30be8fe3cf">◆ </a></span>compute()</h2>
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template<typename MatrixType_ > </div>
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<td class="memname"><a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>& <a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType_ >::compute </td>
<td>(</td>
<td class="paramtype">const <a class="el" href="structEigen_1_1EigenBase.html">EigenBase</a>< InputType > & </td>
<td class="paramname"><em>matrix</em></td><td>)</td>
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<p>Computes the LU decomposition of the given matrix.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tbody><tr><td class="paramname">matrix</td><td>the matrix of which to compute the LU decomposition. It is required to be nonzero.</td></tr>
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<dl class="section return"><dt>Returns</dt><dd>a reference to *this </dd></dl>
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<h2 class="memtitle anchor" id="title10"><span class="permalink"><a href="#a71654e5c60a26407ecccfaa5b34bb0aa">◆ </a></span>determinant()</h2>
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template<typename MatrixType > </div>
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<td class="memname">internal::traits< MatrixType >::Scalar <a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType >::determinant</td>
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<dl class="section return"><dt>Returns</dt><dd>the determinant of the matrix of which *this is the LU decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the LU decomposition has already been computed.</dd></dl>
<dl class="section note"><dt>Note</dt><dd>This is only for square matrices.</dd>
<dd>
For fixed-size matrices of size up to 4, <a class="el" href="classEigen_1_1MatrixBase.html#a7ad8f77004bb956b603bb43fd2e3c061">MatrixBase::determinant()</a> offers optimized paths.</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.</dd></dl>
<dl class="section see"><dt>See also</dt><dd><a class="el" href="classEigen_1_1MatrixBase.html#a7ad8f77004bb956b603bb43fd2e3c061">MatrixBase::determinant()</a> </dd></dl>
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<h2 class="memtitle anchor" id="title11"><span class="permalink"><a href="#a358cec49914ec3cd3707e6b79ae32d0b">◆ </a></span>dimensionOfKernel()</h2>
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<td class="memname"><a class="el" href="structEigen_1_1EigenBase.html#a554f30542cc2316add4b1ea0a492ff02">Index</a> <a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType_ >::dimensionOfKernel </td>
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<dl class="section return"><dt>Returns</dt><dd>the dimension of the kernel of the matrix of which *this is the LU decomposition.</dd></dl>
<dl class="section note"><dt>Note</dt><dd>This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling <a class="el" href="classEigen_1_1FullPivLU.html#abad257b6db0856d8ec52c6072f58f75d">setThreshold(const RealScalar&)</a>. </dd></dl>
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<h2 class="memtitle anchor" id="title12"><span class="permalink"><a href="#aa8cbf984141608e89b503125690d24d4">◆ </a></span>image()</h2>
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<td class="memname">const internal::image_retval<<a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>> <a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType_ >::image </td>
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<dl class="section return"><dt>Returns</dt><dd>the image of the matrix, also called its column-space. The columns of the returned matrix will form a basis of the image (column-space).</dd></dl>
<dl class="params"><dt>Parameters</dt><dd>
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<tbody><tr><td class="paramname">originalMatrix</td><td>the original matrix, of which *this is the LU decomposition. The reason why it is needed to pass it here, is that this allows a large optimization, as otherwise this method would need to reconstruct it from the LU decomposition.</td></tr>
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<dl class="section note"><dt>Note</dt><dd>If the image has dimension zero, then the returned matrix is a column-vector filled with zeros.</dd>
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This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling <a class="el" href="classEigen_1_1FullPivLU.html#abad257b6db0856d8ec52c6072f58f75d">setThreshold(const RealScalar&)</a>.</dd></dl>
<p>Example: </p><div class="fragment"><div class="line"><a class="code" href="group__matrixtypedefs.html#ga84e9fd068879d808012bb6d5dbfecb17">Matrix3d</a> m;</div>
<div class="line">m << 1,1,0,</div>
<div class="line"> 1,3,2,</div>
<div class="line"> 0,1,1;</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">"Notice that the middle column is the sum of the two others, so the "</span></div>
<div class="line"> << <span class="stringliteral">"columns are linearly dependent."</span> << endl;</div>
<div class="line">cout << <span class="stringliteral">"Here is a matrix whose columns have the same span but are linearly independent:"</span></div>
<div class="line"> << endl << m.fullPivLu().image(m) << endl;</div>
<div class="ttc" id="agroup__matrixtypedefs_html_ga84e9fd068879d808012bb6d5dbfecb17"><div class="ttname"><a href="group__matrixtypedefs.html#ga84e9fd068879d808012bb6d5dbfecb17">Eigen::Matrix3d</a></div><div class="ttdeci">Matrix< double, 3, 3 > Matrix3d</div><div class="ttdoc">3×3 matrix of type double.</div><div class="ttdef"><b>Definition:</b> Matrix.h:501</div></div>
</div><!-- fragment --><p> Output: </p><pre class="fragment">Here is the matrix m:
1 1 0
1 3 2
0 1 1
Notice that the middle column is the sum of the two others, so the columns are linearly dependent.
Here is a matrix whose columns have the same span but are linearly independent:
1 1
3 1
1 0
</pre><dl class="section see"><dt>See also</dt><dd><a class="el" href="classEigen_1_1FullPivLU.html#adfc1e27ff60287be5313b5efc3559308">kernel()</a> </dd></dl>
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<h2 class="memtitle anchor" id="title13"><span class="permalink"><a href="#a34afc848d7fb22c7a56a053d3807d2cd">◆ </a></span>inverse()</h2>
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<td class="memname">const <a class="el" href="classEigen_1_1Inverse.html">Inverse</a><<a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>> <a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType_ >::inverse </td>
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<dl class="section return"><dt>Returns</dt><dd>the inverse of the matrix of which *this is the LU decomposition.</dd></dl>
<dl class="section note"><dt>Note</dt><dd>If this matrix is not invertible, the returned matrix has undefined coefficients. Use <a class="el" href="classEigen_1_1FullPivLU.html#a0ee7753645eb31bcbd5faa459168b294">isInvertible()</a> to first determine whether this matrix is invertible.</dd></dl>
<dl class="section see"><dt>See also</dt><dd><a class="el" href="classEigen_1_1MatrixBase.html#a7712eb69e8ea3c8f7b8da1c44dbdeebf">MatrixBase::inverse()</a> </dd></dl>
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<h2 class="memtitle anchor" id="title14"><span class="permalink"><a href="#a90dd33c632ba890175f61eac054bde98">◆ </a></span>isInjective()</h2>
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<dl class="section return"><dt>Returns</dt><dd>true if the matrix of which *this is the LU decomposition represents an injective linear map, i.e. has trivial kernel; false otherwise.</dd></dl>
<dl class="section note"><dt>Note</dt><dd>This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling <a class="el" href="classEigen_1_1FullPivLU.html#abad257b6db0856d8ec52c6072f58f75d">setThreshold(const RealScalar&)</a>. </dd></dl>
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<h2 class="memtitle anchor" id="title15"><span class="permalink"><a href="#a0ee7753645eb31bcbd5faa459168b294">◆ </a></span>isInvertible()</h2>
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<dl class="section return"><dt>Returns</dt><dd>true if the matrix of which *this is the LU decomposition is invertible.</dd></dl>
<dl class="section note"><dt>Note</dt><dd>This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling <a class="el" href="classEigen_1_1FullPivLU.html#abad257b6db0856d8ec52c6072f58f75d">setThreshold(const RealScalar&)</a>. </dd></dl>
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<h2 class="memtitle anchor" id="title16"><span class="permalink"><a href="#af42b9cb6356658b92b2d1006aee73fc4">◆ </a></span>isSurjective()</h2>
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<dl class="section return"><dt>Returns</dt><dd>true if the matrix of which *this is the LU decomposition represents a surjective linear map; false otherwise.</dd></dl>
<dl class="section note"><dt>Note</dt><dd>This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling <a class="el" href="classEigen_1_1FullPivLU.html#abad257b6db0856d8ec52c6072f58f75d">setThreshold(const RealScalar&)</a>. </dd></dl>
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<h2 class="memtitle anchor" id="title17"><span class="permalink"><a href="#adfc1e27ff60287be5313b5efc3559308">◆ </a></span>kernel()</h2>
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<td class="memname">const internal::kernel_retval<<a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>> <a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType_ >::kernel </td>
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<dl class="section return"><dt>Returns</dt><dd>the kernel of the matrix, also called its null-space. The columns of the returned matrix will form a basis of the kernel.</dd></dl>
<dl class="section note"><dt>Note</dt><dd>If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros.</dd>
<dd>
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling <a class="el" href="classEigen_1_1FullPivLU.html#abad257b6db0856d8ec52c6072f58f75d">setThreshold(const RealScalar&)</a>.</dd></dl>
<p>Example: </p><div class="fragment"><div class="line"><a class="code" href="group__matrixtypedefs.html#ga731599f782380312960376c43450eb48">MatrixXf</a> m = <a class="code" href="classEigen_1_1DenseBase.html#ae814abb451b48ed872819192dc188c19">MatrixXf::Random</a>(3,5);</div>
<div class="line">cout << <span class="stringliteral">"Here is the matrix m:"</span> << endl << m << endl;</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga731599f782380312960376c43450eb48">MatrixXf</a> ker = m.fullPivLu().kernel();</div>
<div class="line">cout << <span class="stringliteral">"Here is a matrix whose columns form a basis of the kernel of m:"</span></div>
<div class="line"> << endl << ker << endl;</div>
<div class="line">cout << <span class="stringliteral">"By definition of the kernel, m*ker is zero:"</span></div>
<div class="line"> << endl << m*ker << endl;</div>
<div class="ttc" id="aclassEigen_1_1DenseBase_html_ae814abb451b48ed872819192dc188c19"><div class="ttname"><a href="classEigen_1_1DenseBase.html#ae814abb451b48ed872819192dc188c19">Eigen::DenseBase::Random</a></div><div class="ttdeci">static const RandomReturnType Random()</div><div class="ttdef"><b>Definition:</b> Random.h:114</div></div>
<div class="ttc" id="agroup__matrixtypedefs_html_ga731599f782380312960376c43450eb48"><div class="ttname"><a href="group__matrixtypedefs.html#ga731599f782380312960376c43450eb48">Eigen::MatrixXf</a></div><div class="ttdeci">Matrix< float, Dynamic, Dynamic > MatrixXf</div><div class="ttdoc">Dynamic×Dynamic matrix of type float.</div><div class="ttdef"><b>Definition:</b> Matrix.h:500</div></div>
</div><!-- fragment --><p> Output: </p><pre class="fragment">Here is the matrix m:
0.68 0.597 -0.33 0.108 -0.27
-0.211 0.823 0.536 -0.0452 0.0268
0.566 -0.605 -0.444 0.258 0.904
Here is a matrix whose columns form a basis of the kernel of m:
-0.219 0.763
0.00335 -0.447
0 1
1 0
-0.145 -0.285
By definition of the kernel, m*ker is zero:
7.45e-09 1.49e-08
-1.86e-09 -4.05e-08
0 -2.98e-08
</pre><dl class="section see"><dt>See also</dt><dd><a class="el" href="classEigen_1_1FullPivLU.html#aa8cbf984141608e89b503125690d24d4">image()</a> </dd></dl>
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<h2 class="memtitle anchor" id="title18"><span class="permalink"><a href="#a3e7d7a53f5b7c4ba99013fe171ac5654">◆ </a></span>matrixLU()</h2>
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<td class="memname">const MatrixType& <a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType_ >::matrixLU </td>
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<dl class="section return"><dt>Returns</dt><dd>the LU decomposition matrix: the upper-triangular part is U, the unit-lower-triangular part is L (at least for square matrices; in the non-square case, special care is needed, see the documentation of class <a class="el" href="classEigen_1_1FullPivLU.html" title="LU decomposition of a matrix with complete pivoting, and related features.">FullPivLU</a>).</dd></dl>
<dl class="section see"><dt>See also</dt><dd>matrixL(), matrixU() </dd></dl>
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<h2 class="memtitle anchor" id="title19"><span class="permalink"><a href="#a05cedf8dca6394355ef64c1ea1374b4a">◆ </a></span>maxPivot()</h2>
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<dl class="section return"><dt>Returns</dt><dd>the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of U. </dd></dl>
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<h2 class="memtitle anchor" id="title20"><span class="permalink"><a href="#aad90c46ea08618ae485fce4e5f4677d0">◆ </a></span>nonzeroPivots()</h2>
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<dl class="section return"><dt>Returns</dt><dd>the number of nonzero pivots in the LU decomposition. Here nonzero is meant in the exact sense, not in a fuzzy sense. So that notion isn't really intrinsically interesting, but it is still useful when implementing algorithms.</dd></dl>
<dl class="section see"><dt>See also</dt><dd><a class="el" href="classEigen_1_1FullPivLU.html#a8d31c78a17a70d56ef2d105d6b5efec3">rank()</a> </dd></dl>
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<h2 class="memtitle anchor" id="title21"><span class="permalink"><a href="#a3eb3aa0c37e06ffaf0c07b1eeb0995cc">◆ </a></span>permutationP()</h2>
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<td class="memname">const <a class="el" href="classEigen_1_1PermutationMatrix.html">PermutationPType</a>& <a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType_ >::permutationP </td>
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<dl class="section return"><dt>Returns</dt><dd>the permutation matrix P</dd></dl>
<dl class="section see"><dt>See also</dt><dd><a class="el" href="classEigen_1_1FullPivLU.html#a25b70ffbc88d804981c8874da55e7419">permutationQ()</a> </dd></dl>
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<h2 class="memtitle anchor" id="title22"><span class="permalink"><a href="#a25b70ffbc88d804981c8874da55e7419">◆ </a></span>permutationQ()</h2>
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<dl class="section return"><dt>Returns</dt><dd>the permutation matrix Q</dd></dl>
<dl class="section see"><dt>See also</dt><dd><a class="el" href="classEigen_1_1FullPivLU.html#a3eb3aa0c37e06ffaf0c07b1eeb0995cc">permutationP()</a> </dd></dl>
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<dl class="section return"><dt>Returns</dt><dd>the rank of the matrix of which *this is the LU decomposition.</dd></dl>
<dl class="section note"><dt>Note</dt><dd>This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling <a class="el" href="classEigen_1_1FullPivLU.html#abad257b6db0856d8ec52c6072f58f75d">setThreshold(const RealScalar&)</a>. </dd></dl>
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<h2 class="memtitle anchor" id="title24"><span class="permalink"><a href="#ae39fcfa8d1319472a5b2adfa7a28d9cf">◆ </a></span>rcond()</h2>
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<dl class="section return"><dt>Returns</dt><dd>an estimate of the reciprocal condition number of the matrix of which <code>*this</code> is the LU decomposition. </dd></dl>
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<h2 class="memtitle anchor" id="title25"><span class="permalink"><a href="#a191a4f598b0c192a83ab48984e87ee51">◆ </a></span>reconstructedMatrix()</h2>
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<dl class="section return"><dt>Returns</dt><dd>the matrix represented by the decomposition, i.e., it returns the product: <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0" style=""><nobr><span class="math" id="MathJax-Span-19" style="width: 6.15em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.038em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.94em, 1005.04em, 2.286em, -999.997em); top: -1.929em; left: 0em;"><span class="mrow" id="MathJax-Span-20"><span class="msubsup" id="MathJax-Span-21"><span style="display: inline-block; position: relative; width: 1.818em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.76em, 4.16em, -999.997em); top: -3.978em; left: 0em;"><span class="mi" id="MathJax-Span-22" style="font-family: MathJax_Math-italic;">P<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.12em;"></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.33em; left: 0.823em;"><span class="texatom" id="MathJax-Span-23"><span class="mrow" id="MathJax-Span-24"><span class="mo" id="MathJax-Span-25" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-26" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span><span class="mi" id="MathJax-Span-27" style="font-family: MathJax_Math-italic;">L</span><span class="mi" id="MathJax-Span-28" style="font-family: MathJax_Math-italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.061em;"></span></span><span class="msubsup" id="MathJax-Span-29"><span style="display: inline-block; position: relative; width: 1.759em; height: 0px;"><span style="position: absolute; clip: rect(3.106em, 1000.76em, 4.335em, -999.997em); top: -3.978em; left: 0em;"><span class="mi" id="MathJax-Span-30" style="font-family: MathJax_Math-italic;">Q</span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span><span style="position: absolute; top: -4.33em; left: 0.823em;"><span class="texatom" id="MathJax-Span-31"><span class="mrow" id="MathJax-Span-32"><span class="mo" id="MathJax-Span-33" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-34" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.984em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 1.935em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.282em; border-left: 0px solid; width: 0px; height: 1.432em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-2"> P^{-1} L U Q^{-1} </script>. This function is provided for debug purposes. </dd></dl>
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<h2 class="memtitle anchor" id="title26"><span class="permalink"><a href="#abad257b6db0856d8ec52c6072f58f75d">◆ </a></span>setThreshold() <span class="overload">[1/2]</span></h2>
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<td class="memname"><a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>& <a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType_ >::setThreshold </td>
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<p>Allows to prescribe a threshold to be used by certain methods, such as <a class="el" href="classEigen_1_1FullPivLU.html#a8d31c78a17a70d56ef2d105d6b5efec3">rank()</a>, who need to determine when pivots are to be considered nonzero. This is not used for the LU decomposition itself.</p>
<p>When it needs to get the threshold value, <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library.">Eigen</a> calls <a class="el" href="classEigen_1_1FullPivLU.html#ae2298a7a89749dee7d86f02ccacce0cb">threshold()</a>. By default, this uses a formula to automatically determine a reasonable threshold. Once you have called the present method <a class="el" href="classEigen_1_1FullPivLU.html#abad257b6db0856d8ec52c6072f58f75d">setThreshold(const RealScalar&)</a>, your value is used instead.</p>
<dl class="params"><dt>Parameters</dt><dd>
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<p>A pivot will be considered nonzero if its absolute value is strictly greater than <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0" style=""><nobr><span class="math" id="MathJax-Span-35" style="width: 17.45em; display: inline-block;"><span style="display: inline-block; position: relative; width: 14.289em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.525em, 1014.17em, 2.872em, -999.997em); top: -2.456em; left: 0em;"><span class="mrow" id="MathJax-Span-36"><span class="mo" id="MathJax-Span-37" style="font-family: MathJax_Main;">|</span><span class="mi" id="MathJax-Span-38" style="font-family: MathJax_Math-italic;">p</span><span class="mi" id="MathJax-Span-39" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-40" style="font-family: MathJax_Math-italic;">v</span><span class="mi" id="MathJax-Span-41" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-42" style="font-family: MathJax_Math-italic;">t</span><span class="mo" id="MathJax-Span-43" style="font-family: MathJax_Main;">|</span><span class="mo" id="MathJax-Span-44" style="font-family: MathJax_AMS; padding-left: 0.296em;">⩽</span><span class="mi" id="MathJax-Span-45" style="font-family: MathJax_Math-italic; padding-left: 0.296em;">t</span><span class="mi" id="MathJax-Span-46" style="font-family: MathJax_Math-italic;">h</span><span class="mi" id="MathJax-Span-47" style="font-family: MathJax_Math-italic;">r</span><span class="mi" id="MathJax-Span-48" style="font-family: MathJax_Math-italic;">e</span><span class="mi" id="MathJax-Span-49" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-50" style="font-family: MathJax_Math-italic;">h</span><span class="mi" id="MathJax-Span-51" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-52" style="font-family: MathJax_Math-italic;">l</span><span class="mi" id="MathJax-Span-53" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-54" style="font-family: MathJax_Main; padding-left: 0.237em;">×</span><span class="mo" id="MathJax-Span-55" style="font-family: MathJax_Main; padding-left: 0.237em;">|</span><span class="mi" id="MathJax-Span-56" style="font-family: MathJax_Math-italic;">m</span><span class="mi" id="MathJax-Span-57" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-58" style="font-family: MathJax_Math-italic;">x</span><span class="mi" id="MathJax-Span-59" style="font-family: MathJax_Math-italic;">p</span><span class="mi" id="MathJax-Span-60" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-61" style="font-family: MathJax_Math-italic;">v</span><span class="mi" id="MathJax-Span-62" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-63" style="font-family: MathJax_Math-italic;">t</span><span class="mo" id="MathJax-Span-64" style="font-family: MathJax_Main;">|</span></span><span style="display: inline-block; width: 0px; height: 2.462em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left: 0px solid; width: 0px; height: 1.361em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-3"> \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert </script> where maxpivot is the biggest pivot.</p>
<p>If you want to come back to the default behavior, call <a class="el" href="classEigen_1_1FullPivLU.html#aeafbb0b885cc4c28b53e77988ac5cfe3">setThreshold(Default_t)</a> </p>
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<h2 class="memtitle anchor" id="title27"><span class="permalink"><a href="#aeafbb0b885cc4c28b53e77988ac5cfe3">◆ </a></span>setThreshold() <span class="overload">[2/2]</span></h2>
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<td class="memname"><a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>& <a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType_ >::setThreshold </td>
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<p>Allows to come back to the default behavior, letting <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library.">Eigen</a> use its default formula for determining the threshold.</p>
<p>You should pass the special object Eigen::Default as parameter here. </p><div class="fragment"><div class="line">lu.setThreshold(Eigen::Default); </div>
</div><!-- fragment --><p>See the documentation of <a class="el" href="classEigen_1_1FullPivLU.html#abad257b6db0856d8ec52c6072f58f75d">setThreshold(const RealScalar&)</a>. </p>
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<h2 class="memtitle anchor" id="title28"><span class="permalink"><a href="#a614d3aa28e6af7b2af8630d3e2d022d8">◆ </a></span>solve()</h2>
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<td class="memname">const <a class="el" href="classEigen_1_1Solve.html">Solve</a><<a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>, Rhs> <a class="el" href="classEigen_1_1FullPivLU.html">Eigen::FullPivLU</a>< MatrixType_ >::solve </td>
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<dl class="section return"><dt>Returns</dt><dd>a solution x to the equation Ax=b, where A is the matrix of which *this is the LU decomposition.</dd></dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tbody><tr><td class="paramname">b</td><td>the right-hand-side of the equation to solve. Can be a vector or a matrix, the only requirement in order for the equation to make sense is that b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.</td></tr>
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<dl class="section return"><dt>Returns</dt><dd>a solution.</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#ae8443357b808cd393be1b51974213f9c">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. If you need a complete analysis of the space of solutions, take the one solution obtained by this method and add to it elements of the kernel, as determined by <a class="el" href="classEigen_1_1FullPivLU.html#adfc1e27ff60287be5313b5efc3559308">kernel()</a>.</p>
<p>Example: </p><div class="fragment"><div class="line">Matrix<float,2,3> m = <a class="code" href="classEigen_1_1DenseBase.html#ae814abb451b48ed872819192dc188c19">Matrix<float,2,3>::Random</a>();</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga36b8989b6aa63020139fc36bae6979e0">Matrix2f</a> y = <a class="code" href="classEigen_1_1DenseBase.html#ae814abb451b48ed872819192dc188c19">Matrix2f::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">Matrix<float,3,2> x = m.fullPivLu().solve(y);</div>
<div class="line"><span class="keywordflow">if</span>((m*x).isApprox(y))</div>
<div class="line">{</div>
<div class="line"> cout << <span class="stringliteral">"Here is a solution x to the equation mx=y:"</span> << endl << x << endl;</div>
<div class="line">}</div>
<div class="line"><span class="keywordflow">else</span></div>
<div class="line"> cout << <span class="stringliteral">"The equation mx=y does not have any solution."</span> << endl;</div>
<div class="ttc" id="agroup__matrixtypedefs_html_ga36b8989b6aa63020139fc36bae6979e0"><div class="ttname"><a href="group__matrixtypedefs.html#ga36b8989b6aa63020139fc36bae6979e0">Eigen::Matrix2f</a></div><div class="ttdeci">Matrix< float, 2, 2 > Matrix2f</div><div class="ttdoc">2×2 matrix of type float.</div><div class="ttdef"><b>Definition:</b> Matrix.h:500</div></div>
</div><!-- fragment --><p> Output: </p><pre class="fragment">Here is the matrix m:
0.68 0.566 0.823
-0.211 0.597 -0.605
Here is the matrix y:
-0.33 -0.444
0.536 0.108
Here is a solution x to the equation mx=y:
0 0
0.291 -0.216
-0.6 -0.391
</pre><dl class="section see"><dt>See also</dt><dd>TriangularView::solve(), <a class="el" href="classEigen_1_1FullPivLU.html#adfc1e27ff60287be5313b5efc3559308">kernel()</a>, <a class="el" href="classEigen_1_1FullPivLU.html#a34afc848d7fb22c7a56a053d3807d2cd">inverse()</a> </dd></dl>
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<h2 class="memtitle anchor" id="title29"><span class="permalink"><a href="#ae2298a7a89749dee7d86f02ccacce0cb">◆ </a></span>threshold()</h2>
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<p>Returns the threshold that will be used by certain methods such as <a class="el" href="classEigen_1_1FullPivLU.html#a8d31c78a17a70d56ef2d105d6b5efec3">rank()</a>.</p>
<p>See the documentation of <a class="el" href="classEigen_1_1FullPivLU.html#abad257b6db0856d8ec52c6072f58f75d">setThreshold(const RealScalar&)</a>. </p>
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<hr>The documentation for this class was generated from the following file:<ul>
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