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<a href="classEigen_1_1FullPivLU-members.html">List of all members</a> |
<a href="#pub-methods">Public Member Functions</a> </div>
<div class="headertitle">
<div class="title">FullPivLU< MatrixType > Class Template Reference<div class="ingroups"><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">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="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 LU decomposition</td></tr>
</table>
</dd>
</dl>
<p>This class represents a LU decomposition of any matrix, with complete pivoting: the matrix A is decomposed as A = PLUQ 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#ad69664a62ab4d3026566d0d4a261b187">matrixLU()</a>, <a class="el" href="classEigen_1_1FullPivLU.html#a38416f9985b9c7ad9dc3bd355479dd67">permutationP()</a>, <a class="el" href="classEigen_1_1FullPivLU.html#ae85e85c3d1488b5882a6ccd63678a4d1">permutationQ()</a>.</p>
<p>As an exemple, 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#ggab59c1bec446b10af208f977a871d910bae70afef0d3ff7aca74e17e85ff6c9f2e">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><!-- 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><dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1MatrixBase.html#a0844e94f8f95ae01a2cd88dbbf5cbf91">MatrixBase::fullPivLu()</a>, <a class="el" href="classEigen_1_1MatrixBase.html#ad63cea11a4bf220298dce4489a1704c7">MatrixBase::determinant()</a>, <a class="el" href="classEigen_1_1MatrixBase.html#aa2834da4c855fa35fed8c4030f79f9da">MatrixBase::inverse()</a> </dd></dl>
</div><table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="pub-methods"></a>
Public Member Functions</h2></td></tr>
<tr class="memitem:ac1c60e27ca149bb599662fb554a0949a"><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#ac1c60e27ca149bb599662fb554a0949a">compute</a> (const MatrixType &matrix)</td></tr>
<tr class="separator:ac1c60e27ca149bb599662fb554a0949a"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aa47f041dae554fe1f135e2794ae914a7"><td class="memItemLeft" align="right" valign="top">internal::traits< MatrixType ><br class="typebreak"/>
::Scalar </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#aa47f041dae554fe1f135e2794ae914a7">determinant</a> () const </td></tr>
<tr class="separator:aa47f041dae554fe1f135e2794ae914a7"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a7c6323871c4f080fc6e2d3ad7fc607fc"><td class="memItemLeft" align="right" valign="top">Index </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a7c6323871c4f080fc6e2d3ad7fc607fc">dimensionOfKernel</a> () const </td></tr>
<tr class="separator:a7c6323871c4f080fc6e2d3ad7fc607fc"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a10dec2fa1767ac0712af1efa732b2046"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a10dec2fa1767ac0712af1efa732b2046">FullPivLU</a> ()</td></tr>
<tr class="memdesc:a10dec2fa1767ac0712af1efa732b2046"><td class="mdescLeft"> </td><td class="mdescRight">Default Constructor. <a href="#a10dec2fa1767ac0712af1efa732b2046">More...</a><br/></td></tr>
<tr class="separator:a10dec2fa1767ac0712af1efa732b2046"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ab6c78ecd953ba165ef421fa67c05753a"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#ab6c78ecd953ba165ef421fa67c05753a">FullPivLU</a> (Index rows, Index cols)</td></tr>
<tr class="memdesc:ab6c78ecd953ba165ef421fa67c05753a"><td class="mdescLeft"> </td><td class="mdescRight">Default Constructor with memory preallocation. <a href="#ab6c78ecd953ba165ef421fa67c05753a">More...</a><br/></td></tr>
<tr class="separator:ab6c78ecd953ba165ef421fa67c05753a"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ab66f95f8f6b6455a0a38d95486330808"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#ab66f95f8f6b6455a0a38d95486330808">FullPivLU</a> (const MatrixType &matrix)</td></tr>
<tr class="separator:ab66f95f8f6b6455a0a38d95486330808"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ad854843c6dc601252dc107e7b29133e9"><td class="memItemLeft" align="right" valign="top">const internal::image_retval<br class="typebreak"/>
< <a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a> > </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#ad854843c6dc601252dc107e7b29133e9">image</a> (const MatrixType &originalMatrix) const </td></tr>
<tr class="separator:ad854843c6dc601252dc107e7b29133e9"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a5ca20e3802e96fb14c2be37039afcae9"><td class="memItemLeft" align="right" valign="top">const internal::solve_retval<br class="typebreak"/>
< <a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>, typename <br class="typebreak"/>
MatrixType::IdentityReturnType > </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a5ca20e3802e96fb14c2be37039afcae9">inverse</a> () const </td></tr>
<tr class="separator:a5ca20e3802e96fb14c2be37039afcae9"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a1e119085e53eca65e9ba15451c102d40"><td class="memItemLeft" align="right" valign="top">bool </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a1e119085e53eca65e9ba15451c102d40">isInjective</a> () const </td></tr>
<tr class="separator:a1e119085e53eca65e9ba15451c102d40"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ab60c7d993c9eba31668fb8886d621094"><td class="memItemLeft" align="right" valign="top">bool </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#ab60c7d993c9eba31668fb8886d621094">isInvertible</a> () const </td></tr>
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<tr class="memitem:a5968b9ca46303b3cc7250e7b120ab7e6"><td class="memItemLeft" align="right" valign="top">bool </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a5968b9ca46303b3cc7250e7b120ab7e6">isSurjective</a> () const </td></tr>
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<tr class="memitem:a6e8f1d2fcbd86d3dc5a8a013b6e7200a"><td class="memItemLeft" align="right" valign="top">const internal::kernel_retval<br class="typebreak"/>
< <a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a> > </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a6e8f1d2fcbd86d3dc5a8a013b6e7200a">kernel</a> () const </td></tr>
<tr class="separator:a6e8f1d2fcbd86d3dc5a8a013b6e7200a"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ad69664a62ab4d3026566d0d4a261b187"><td class="memItemLeft" align="right" valign="top">const MatrixType & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#ad69664a62ab4d3026566d0d4a261b187">matrixLU</a> () const </td></tr>
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<tr class="memitem:a067e9d4143ce0558fc684b736128a4ed"><td class="memItemLeft" align="right" valign="top">RealScalar </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a067e9d4143ce0558fc684b736128a4ed">maxPivot</a> () const </td></tr>
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<tr class="memitem:a6dda9285f13dec9f49e9c17229a89988"><td class="memItemLeft" align="right" valign="top">Index </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a6dda9285f13dec9f49e9c17229a89988">nonzeroPivots</a> () const </td></tr>
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<tr class="memitem:a38416f9985b9c7ad9dc3bd355479dd67"><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#a38416f9985b9c7ad9dc3bd355479dd67">permutationP</a> () const </td></tr>
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<tr class="memitem:ae85e85c3d1488b5882a6ccd63678a4d1"><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#ae85e85c3d1488b5882a6ccd63678a4d1">permutationQ</a> () const </td></tr>
<tr class="separator:ae85e85c3d1488b5882a6ccd63678a4d1"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a363d1c09d77f09d6ea2d2789776e7be3"><td class="memItemLeft" align="right" valign="top">Index </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#a363d1c09d77f09d6ea2d2789776e7be3">rank</a> () const </td></tr>
<tr class="separator:a363d1c09d77f09d6ea2d2789776e7be3"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ac36925ac693435a090efee1cb5d6d16a"><td class="memItemLeft" align="right" valign="top">MatrixType </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#ac36925ac693435a090efee1cb5d6d16a">reconstructedMatrix</a> () const </td></tr>
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<tr class="memitem:ae47399e3cf7943075ce18ef89fe17f21"><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#ae47399e3cf7943075ce18ef89fe17f21">setThreshold</a> (const RealScalar &<a class="el" href="classEigen_1_1FullPivLU.html#aa5a87faaa92a3081045d1f934e292ef0">threshold</a>)</td></tr>
<tr class="separator:ae47399e3cf7943075ce18ef89fe17f21"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a02b74b9b823e060825443fd82cd25fd4"><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#a02b74b9b823e060825443fd82cd25fd4">setThreshold</a> (Default_t)</td></tr>
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<tr class="memitem:ac1a642d728c059c7625863f126e2e718"><td class="memTemplParams" colspan="2">template<typename Rhs > </td></tr>
<tr class="memitem:ac1a642d728c059c7625863f126e2e718"><td class="memTemplItemLeft" align="right" valign="top">const internal::solve_retval<br class="typebreak"/>
< <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#ac1a642d728c059c7625863f126e2e718">solve</a> (const <a class="el" href="classEigen_1_1MatrixBase.html">MatrixBase</a>< Rhs > &b) const </td></tr>
<tr class="separator:ac1a642d728c059c7625863f126e2e718"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aa5a87faaa92a3081045d1f934e292ef0"><td class="memItemLeft" align="right" valign="top">RealScalar </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1FullPivLU.html#aa5a87faaa92a3081045d1f934e292ef0">threshold</a> () const </td></tr>
<tr class="separator:aa5a87faaa92a3081045d1f934e292ef0"><td class="memSeparator" colspan="2"> </td></tr>
</table>
<h2 class="groupheader">Constructor & Destructor Documentation</h2>
<a class="anchor" id="a10dec2fa1767ac0712af1efa732b2046"></a>
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<div class="memproto">
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<td class="memname"><a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a> </td>
<td>(</td>
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<p>Default Constructor. </p>
<p>The default constructor is useful in cases in which the user intends to perform decompositions via LU::compute(const MatrixType&). </p>
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<td class="paramtype">Index </td>
<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_1FullPivLU.html#a10dec2fa1767ac0712af1efa732b2046" title="Default Constructor. ">FullPivLU()</a> </dd></dl>
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<p>Constructor.</p>
<dl class="params"><dt>Parameters</dt><dd>
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<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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<p>References <a class="el" href="classEigen_1_1FullPivLU.html#ac1c60e27ca149bb599662fb554a0949a">FullPivLU< MatrixType >::compute()</a>.</p>
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<h2 class="groupheader">Member Function Documentation</h2>
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<td class="memname"><a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>< MatrixType > & compute </td>
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<p>Computes the LU decomposition of the given matrix.</p>
<dl class="params"><dt>Parameters</dt><dd>
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<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>
<p>Referenced by <a class="el" href="classEigen_1_1FullPivLU.html#ab66f95f8f6b6455a0a38d95486330808">FullPivLU< MatrixType >::FullPivLU()</a>.</p>
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<td class="memname">internal::traits< MatrixType >::Scalar 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#ad63cea11a4bf220298dce4489a1704c7">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#ad63cea11a4bf220298dce4489a1704c7">MatrixBase::determinant()</a> </dd></dl>
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<td class="memname">Index 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#ae47399e3cf7943075ce18ef89fe17f21">setThreshold(const RealScalar&)</a>. </dd></dl>
<p>References <a class="el" href="classEigen_1_1FullPivLU.html#a363d1c09d77f09d6ea2d2789776e7be3">FullPivLU< MatrixType >::rank()</a>.</p>
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<td class="memname">const internal::image_retval<<a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>> 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 kernel.</dd></dl>
<dl class="params"><dt>Parameters</dt><dd>
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<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>
<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#ae47399e3cf7943075ce18ef89fe17f21">setThreshold(const RealScalar&)</a>.</dd></dl>
<p>Example: </p>
<div class="fragment"><div class="line"><a class="code" href="group__matrixtypedefs.html#ga45a14b423c687c3e2e8325f148e27bf3">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><!-- 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#a6e8f1d2fcbd86d3dc5a8a013b6e7200a">kernel()</a> </dd></dl>
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<td class="memname">const internal::solve_retval<<a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>,typename MatrixType::IdentityReturnType> 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#ab60c7d993c9eba31668fb8886d621094">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#aa2834da4c855fa35fed8c4030f79f9da">MatrixBase::inverse()</a> </dd></dl>
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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#ae47399e3cf7943075ce18ef89fe17f21">setThreshold(const RealScalar&)</a>. </dd></dl>
<p>References <a class="el" href="classEigen_1_1FullPivLU.html#a363d1c09d77f09d6ea2d2789776e7be3">FullPivLU< MatrixType >::rank()</a>.</p>
<p>Referenced by <a class="el" href="classEigen_1_1FullPivLU.html#ab60c7d993c9eba31668fb8886d621094">FullPivLU< MatrixType >::isInvertible()</a>.</p>
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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#ae47399e3cf7943075ce18ef89fe17f21">setThreshold(const RealScalar&)</a>. </dd></dl>
<p>References <a class="el" href="classEigen_1_1FullPivLU.html#a1e119085e53eca65e9ba15451c102d40">FullPivLU< MatrixType >::isInjective()</a>.</p>
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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#ae47399e3cf7943075ce18ef89fe17f21">setThreshold(const RealScalar&)</a>. </dd></dl>
<p>References <a class="el" href="classEigen_1_1FullPivLU.html#a363d1c09d77f09d6ea2d2789776e7be3">FullPivLU< MatrixType >::rank()</a>.</p>
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<td class="memname">const internal::kernel_retval<<a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>> 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#ae47399e3cf7943075ce18ef89fe17f21">setThreshold(const RealScalar&)</a>.</dd></dl>
<p>Example: </p>
<div class="fragment"><div class="line"><a class="code" href="group__matrixtypedefs.html#gabab09c32e96cfa9829a88400627af162">MatrixXf</a> m = <a class="code" href="classEigen_1_1DenseBase.html#a8e759dafdd9ecc446d397b7f5435f60a">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#gabab09c32e96cfa9829a88400627af162">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><!-- 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:
-1.12e-08 1.49e-08
-1.4e-09 -4.05e-08
1.49e-08 -2.98e-08
</pre><dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1FullPivLU.html#ad854843c6dc601252dc107e7b29133e9">image()</a> </dd></dl>
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<td class="memname">const 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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<td class="memname">RealScalar maxPivot </td>
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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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<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#a363d1c09d77f09d6ea2d2789776e7be3">rank()</a> </dd></dl>
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<td class="memname">const <a class="el" href="classEigen_1_1PermutationMatrix.html">PermutationPType</a>& 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#ae85e85c3d1488b5882a6ccd63678a4d1">permutationQ()</a> </dd></dl>
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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#a38416f9985b9c7ad9dc3bd355479dd67">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#ae47399e3cf7943075ce18ef89fe17f21">setThreshold(const RealScalar&)</a>. </dd></dl>
<p>References <a class="el" href="classEigen_1_1FullPivLU.html#aa5a87faaa92a3081045d1f934e292ef0">FullPivLU< MatrixType >::threshold()</a>.</p>
<p>Referenced by <a class="el" href="classEigen_1_1FullPivLU.html#a7c6323871c4f080fc6e2d3ad7fc607fc">FullPivLU< MatrixType >::dimensionOfKernel()</a>, <a class="el" href="classEigen_1_1FullPivLU.html#a1e119085e53eca65e9ba15451c102d40">FullPivLU< MatrixType >::isInjective()</a>, and <a class="el" href="classEigen_1_1FullPivLU.html#a5968b9ca46303b3cc7250e7b120ab7e6">FullPivLU< MatrixType >::isSurjective()</a>.</p>
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<td class="memname">MatrixType reconstructedMatrix </td>
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<dl class="section return"><dt>Returns</dt><dd>the matrix represented by the decomposition, i.e., it returns the product: P^{-1} L U Q^{-1}. This function is provided for debug purpose. </dd></dl>
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<td class="memname"><a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>& setThreshold </td>
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<td class="paramtype">const RealScalar & </td>
<td class="paramname"><em>threshold</em></td><td>)</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#a363d1c09d77f09d6ea2d2789776e7be3">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#aa5a87faaa92a3081045d1f934e292ef0">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#ae47399e3cf7943075ce18ef89fe17f21">setThreshold(const RealScalar&)</a>, your value is used instead.</p>
<dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramname">threshold</td><td>The new value to use as the threshold.</td></tr>
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<p>A pivot will be considered nonzero if its absolute value is strictly greater than <img class="formulaInl" alt="$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert $" src="form_156.png"/> 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#a02b74b9b823e060825443fd82cd25fd4">setThreshold(Default_t)</a> </p>
<p>References <a class="el" href="classEigen_1_1FullPivLU.html#aa5a87faaa92a3081045d1f934e292ef0">FullPivLU< MatrixType >::threshold()</a>.</p>
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<td class="memname"><a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>& 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#ae47399e3cf7943075ce18ef89fe17f21">setThreshold(const RealScalar&)</a>. </p>
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<td class="memname">const internal::solve_retval<<a class="el" href="classEigen_1_1FullPivLU.html">FullPivLU</a>, Rhs> solve </td>
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<td class="paramname"><em>b</em></td><td>)</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>
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<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#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. 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#a6e8f1d2fcbd86d3dc5a8a013b6e7200a">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#a8e759dafdd9ecc446d397b7f5435f60a">Matrix<float,2,3>::Random</a>();</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga535a919504bb3bc463b8995c196c1eed">Matrix2f</a> y = <a class="code" href="classEigen_1_1DenseBase.html#a8e759dafdd9ecc446d397b7f5435f60a">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><!-- 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#a6e8f1d2fcbd86d3dc5a8a013b6e7200a">kernel()</a>, <a class="el" href="classEigen_1_1FullPivLU.html#a5ca20e3802e96fb14c2be37039afcae9">inverse()</a> </dd></dl>
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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#a363d1c09d77f09d6ea2d2789776e7be3">rank()</a>.</p>
<p>See the documentation of <a class="el" href="classEigen_1_1FullPivLU.html#ae47399e3cf7943075ce18ef89fe17f21">setThreshold(const RealScalar&)</a>. </p>
<p>Referenced by <a class="el" href="classEigen_1_1FullPivLU.html#a363d1c09d77f09d6ea2d2789776e7be3">FullPivLU< MatrixType >::rank()</a>, and <a class="el" href="classEigen_1_1FullPivLU.html#ae47399e3cf7943075ce18ef89fe17f21">FullPivLU< MatrixType >::setThreshold()</a>.</p>
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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="FullPivLU_8h_source.html">FullPivLU.h</a></li>
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