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<div class="title">JacobiSVD< _MatrixType, QRPreconditioner > Class Template Reference<div class="ingroups"><a class="el" href="group__SVD__Module.html">SVD 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, int QRPreconditioner><br/>
class Eigen::JacobiSVD< _MatrixType, QRPreconditioner ></h3>
<p>Two-sided Jacobi SVD decomposition of a rectangular matrix. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">MatrixType</td><td>the type of the matrix of which we are computing the SVD decomposition </td></tr>
<tr><td class="paramname">QRPreconditioner</td><td>this optional parameter allows to specify the type of QR decomposition that will be used internally for the R-SVD step for non-square matrices. See discussion of possible values below.</td></tr>
</table>
</dd>
</dl>
<p>SVD decomposition consists in decomposing any n-by-p matrix <em>A</em> as a product </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ A = U S V^* \]" src="form_93.png"/>
</p>
<p> where <em>U</em> is a n-by-n unitary, <em>V</em> is a p-by-p unitary, and <em>S</em> is a n-by-p real positive matrix which is zero outside of its main diagonal; the diagonal entries of S are known as the <em>singular</em> <em>values</em> of <em>A</em> and the columns of <em>U</em> and <em>V</em> are known as the left and right <em>singular</em> <em>vectors</em> of <em>A</em> respectively.</p>
<p>Singular values are always sorted in decreasing order.</p>
<p>This <a class="el" href="classEigen_1_1JacobiSVD.html" title="Two-sided Jacobi SVD decomposition of a rectangular matrix. ">JacobiSVD</a> decomposition computes only the singular values by default. If you want <em>U</em> or <em>V</em>, you need to ask for them explicitly.</p>
<p>You can ask for only <em>thin</em> <em>U</em> or <em>V</em> to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting <em>m</em> be the smaller value among <em>n</em> and <em>p</em>, there are only <em>m</em> singular vectors; the remaining columns of <em>U</em> and <em>V</em> do not correspond to actual singular vectors. Asking for <em>thin</em> <em>U</em> or <em>V</em> means asking for only their <em>m</em> first columns to be formed. So <em>U</em> is then a n-by-m matrix, and <em>V</em> is then a p-by-m matrix. Notice that thin <em>U</em> and <em>V</em> are all you need for (least squares) solving.</p>
<p>Here's an example demonstrating basic usage: </p>
<div class="fragment"><div class="line">MatrixXf m = MatrixXf::Random(3,2);</div>
<div class="line">cout << <span class="stringliteral">"Here is the matrix m:"</span> << endl << m << endl;</div>
<div class="line">JacobiSVD<MatrixXf> svd(m, ComputeThinU | ComputeThinV);</div>
<div class="line">cout << <span class="stringliteral">"Its singular values are:"</span> << endl << svd.singularValues() << endl;</div>
<div class="line">cout << <span class="stringliteral">"Its left singular vectors are the columns of the thin U matrix:"</span> << endl << svd.matrixU() << endl;</div>
<div class="line">cout << <span class="stringliteral">"Its right singular vectors are the columns of the thin V matrix:"</span> << endl << svd.matrixV() << endl;</div>
<div class="line">Vector3f rhs(1, 0, 0);</div>
<div class="line">cout << <span class="stringliteral">"Now consider this rhs vector:"</span> << endl << rhs << endl;</div>
<div class="line">cout << <span class="stringliteral">"A least-squares solution of m*x = rhs is:"</span> << endl << svd.solve(rhs) << endl;</div>
</div><!-- fragment --><p> Output: </p>
<pre class="fragment">Here is the matrix m:
0.68 0.597
-0.211 0.823
0.566 -0.605
Its singular values are:
1.19
0.899
Its left singular vectors are the columns of the thin U matrix:
0.388 0.866
0.712 -0.0634
-0.586 0.496
Its right singular vectors are the columns of the thin V matrix:
-0.183 0.983
0.983 0.183
Now consider this rhs vector:
1
0
0
A least-squares solution of m*x = rhs is:
0.888
0.496
</pre><p>This <a class="el" href="classEigen_1_1JacobiSVD.html" title="Two-sided Jacobi SVD decomposition of a rectangular matrix. ">JacobiSVD</a> class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than bidiagonalizing SVD algorithms for large square matrices; however its complexity is still <img class="formulaInl" alt="$ O(n^2p) $" src="form_94.png"/> where <em>n</em> is the smaller dimension and <em>p</em> is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms. In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension.</p>
<p>If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to terminate in finite (and reasonable) time.</p>
<p>The possible values for QRPreconditioner are: </p>
<ul>
<li>ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR. </li>
<li>FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR. Contrary to other QRs, it doesn't allow computing thin unitaries. </li>
<li>HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR. This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive process is more reliable than the optimized bidiagonal SVD iterations. </li>
<li>NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing <a class="el" href="classEigen_1_1JacobiSVD.html" title="Two-sided Jacobi SVD decomposition of a rectangular matrix. ">JacobiSVD</a> decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in faster compilation and smaller executable code. It won't significantly speed up computation, since <a class="el" href="classEigen_1_1JacobiSVD.html" title="Two-sided Jacobi SVD decomposition of a rectangular matrix. ">JacobiSVD</a> is always checking if QR preconditioning is needed before applying it anyway.</li>
</ul>
<dl class="section see"><dt>See Also</dt><dd>MatrixBase::jacobiSvd() </dd></dl>
</div><div id="dynsection-0" onclick="return toggleVisibility(this)" class="dynheader closed" style="cursor:pointer;">
<img id="dynsection-0-trigger" src="closed.png" alt="+"/> Inheritance diagram for JacobiSVD< _MatrixType, QRPreconditioner >:</div>
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<map name="JacobiSVD_3_01__MatrixType_00_01QRPreconditioner_01_4_inherit__map" id="JacobiSVD_3_01__MatrixType_00_01QRPreconditioner_01_4_inherit__map">
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<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:a3b2bfdc0a8dd672390fb4ba22f4ef434"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1SVDBase.html">SVDBase</a>< MatrixType > & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1JacobiSVD.html#a3b2bfdc0a8dd672390fb4ba22f4ef434">compute</a> (const MatrixType &matrix, unsigned int computationOptions)</td></tr>
<tr class="memdesc:a3b2bfdc0a8dd672390fb4ba22f4ef434"><td class="mdescLeft"> </td><td class="mdescRight">Method performing the decomposition of given matrix using custom options. <a href="#a3b2bfdc0a8dd672390fb4ba22f4ef434">More...</a><br/></td></tr>
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<tr class="memitem:a26e02670d0a94c92ab41c2bc7f70e781"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1SVDBase.html">SVDBase</a>< MatrixType > & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1JacobiSVD.html#a26e02670d0a94c92ab41c2bc7f70e781">compute</a> (const MatrixType &matrix)</td></tr>
<tr class="memdesc:a26e02670d0a94c92ab41c2bc7f70e781"><td class="mdescLeft"> </td><td class="mdescRight">Method performing the decomposition of given matrix using current options. <a href="#a26e02670d0a94c92ab41c2bc7f70e781">More...</a><br/></td></tr>
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<tr class="memitem:a1ffab6aab715fe0918a841611a95e9aa"><td class="memItemLeft" align="right" valign="top">bool </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1SVDBase.html#a1ffab6aab715fe0918a841611a95e9aa">computeU</a> () const </td></tr>
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<tr class="memitem:a92e99646eefbeb071ef220841555a703"><td class="memItemLeft" align="right" valign="top">bool </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1SVDBase.html#a92e99646eefbeb071ef220841555a703">computeV</a> () const </td></tr>
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<tr class="memitem:a0e963136a69da877ff06f27e7b746351"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1JacobiSVD.html#a0e963136a69da877ff06f27e7b746351">JacobiSVD</a> ()</td></tr>
<tr class="memdesc:a0e963136a69da877ff06f27e7b746351"><td class="mdescLeft"> </td><td class="mdescRight">Default Constructor. <a href="#a0e963136a69da877ff06f27e7b746351">More...</a><br/></td></tr>
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<tr class="memitem:a18cfaad45164fc79a0b5e65c194d049d"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1JacobiSVD.html#a18cfaad45164fc79a0b5e65c194d049d">JacobiSVD</a> (Index rows, Index cols, unsigned int computationOptions=0)</td></tr>
<tr class="memdesc:a18cfaad45164fc79a0b5e65c194d049d"><td class="mdescLeft"> </td><td class="mdescRight">Default Constructor with memory preallocation. <a href="#a18cfaad45164fc79a0b5e65c194d049d">More...</a><br/></td></tr>
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<tr class="memitem:af7d98465f0e886d96423857591a34b26"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1JacobiSVD.html#af7d98465f0e886d96423857591a34b26">JacobiSVD</a> (const MatrixType &matrix, unsigned int computationOptions=0)</td></tr>
<tr class="memdesc:af7d98465f0e886d96423857591a34b26"><td class="mdescLeft"> </td><td class="mdescRight">Constructor performing the decomposition of given matrix. <a href="#af7d98465f0e886d96423857591a34b26">More...</a><br/></td></tr>
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<tr class="memitem:a49e16a4adf4fe58a2d65a5e5a31e7654"><td class="memItemLeft" align="right" valign="top">const MatrixUType & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1SVDBase.html#a49e16a4adf4fe58a2d65a5e5a31e7654">matrixU</a> () const </td></tr>
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<tr class="memitem:ae5158ab7ca44a705c2a3b56ec926b42a"><td class="memItemLeft" align="right" valign="top">const MatrixVType & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1SVDBase.html#ae5158ab7ca44a705c2a3b56ec926b42a">matrixV</a> () const </td></tr>
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<tr class="memitem:aa468765ea9b7f0e92380fa206e6498bd"><td class="memItemLeft" align="right" valign="top">Index </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1SVDBase.html#aa468765ea9b7f0e92380fa206e6498bd">nonzeroSingularValues</a> () const </td></tr>
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<tr class="memitem:a48d4068b97dfbb83d62599e56e26797a"><td class="memItemLeft" align="right" valign="top">const SingularValuesType & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1SVDBase.html#a48d4068b97dfbb83d62599e56e26797a">singularValues</a> () const </td></tr>
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<tr class="memitem:ae86e342cd51b067b08f8de8bae77537f"><td class="memTemplParams" colspan="2">template<typename Rhs > </td></tr>
<tr class="memitem:ae86e342cd51b067b08f8de8bae77537f"><td class="memTemplItemLeft" align="right" valign="top">const internal::solve_retval<br class="typebreak"/>
< <a class="el" href="classEigen_1_1JacobiSVD.html">JacobiSVD</a>, Rhs > </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="classEigen_1_1JacobiSVD.html#ae86e342cd51b067b08f8de8bae77537f">solve</a> (const MatrixBase< Rhs > &b) const </td></tr>
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<h2 class="groupheader">Constructor & Destructor Documentation</h2>
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<p>Default Constructor. </p>
<p>The default constructor is useful in cases in which the user intends to perform decompositions via <a class="el" href="classEigen_1_1JacobiSVD.html#a26e02670d0a94c92ab41c2bc7f70e781" title="Method performing the decomposition of given matrix using current options. ">JacobiSVD::compute(const MatrixType&)</a>. </p>
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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 size. </p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1JacobiSVD.html#a0e963136a69da877ff06f27e7b746351" title="Default Constructor. ">JacobiSVD()</a> </dd></dl>
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<p>Constructor performing the decomposition of given matrix. </p>
<dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramname">matrix</td><td>the matrix to decompose </td></tr>
<tr><td class="paramname">computationOptions</td><td>optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU, #ComputeFullV, #ComputeThinV.</td></tr>
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<p>Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not available with the (non-default) FullPivHouseholderQR preconditioner. </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_1SVDBase.html">SVDBase</a>< MatrixType > & compute </td>
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<td class="paramtype">const MatrixType & </td>
<td class="paramname"><em>matrix</em>, </td>
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<td class="paramtype">unsigned int </td>
<td class="paramname"><em>computationOptions</em> </td>
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<p>Method performing the decomposition of given matrix using custom options. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">matrix</td><td>the matrix to decompose </td></tr>
<tr><td class="paramname">computationOptions</td><td>optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU, #ComputeFullV, #ComputeThinV.</td></tr>
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<p>Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not available with the (non-default) FullPivHouseholderQR preconditioner. </p>
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<td class="memname"><a class="el" href="classEigen_1_1SVDBase.html">SVDBase</a><MatrixType>& compute </td>
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<td class="paramtype">const MatrixType & </td>
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<p>Method performing the decomposition of given matrix using current options. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">matrix</td><td>the matrix to decompose</td></tr>
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<p>This method uses the current <em>computationOptions</em>, as already passed to the constructor or to <a class="el" href="classEigen_1_1JacobiSVD.html#a3b2bfdc0a8dd672390fb4ba22f4ef434" title="Method performing the decomposition of given matrix using custom options. ">compute(const MatrixType&, unsigned int)</a>. </p>
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<td class="memname">bool computeU </td>
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<td class="paramname"></td><td>)</td>
<td> const</td>
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<dl class="section return"><dt>Returns</dt><dd>true if <em>U</em> (full or thin) is asked for in this SVD decomposition </dd></dl>
<p>Referenced by <a class="el" href="classEigen_1_1SVDBase.html#a49e16a4adf4fe58a2d65a5e5a31e7654">SVDBase< _MatrixType >::matrixU()</a>.</p>
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<td class="memname">bool computeV </td>
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<dl class="section return"><dt>Returns</dt><dd>true if <em>V</em> (full or thin) is asked for in this SVD decomposition </dd></dl>
<p>Referenced by <a class="el" href="classEigen_1_1SVDBase.html#ae5158ab7ca44a705c2a3b56ec926b42a">SVDBase< _MatrixType >::matrixV()</a>.</p>
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<td class="memname">const MatrixUType& matrixU </td>
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<td> const</td>
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<dl class="section return"><dt>Returns</dt><dd>the <em>U</em> matrix.</dd></dl>
<p>For the <a class="el" href="classEigen_1_1SVDBase.html" title="Mother class of SVD classes algorithms. ">SVDBase</a> decomposition of a n-by-p matrix, letting <em>m</em> be the minimum of <em>n</em> and <em>p</em>, the U matrix is n-by-n if you asked for #ComputeFullU, and is n-by-m if you asked for #ComputeThinU.</p>
<p>The <em>m</em> first columns of <em>U</em> are the left singular vectors of the matrix being decomposed.</p>
<p>This method asserts that you asked for <em>U</em> to be computed. </p>
<p>References <a class="el" href="classEigen_1_1SVDBase.html#a1ffab6aab715fe0918a841611a95e9aa">SVDBase< _MatrixType >::computeU()</a>.</p>
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<td class="memname">const MatrixVType& matrixV </td>
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<td> const</td>
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<dl class="section return"><dt>Returns</dt><dd>the <em>V</em> matrix.</dd></dl>
<p>For the SVD decomposition of a n-by-p matrix, letting <em>m</em> be the minimum of <em>n</em> and <em>p</em>, the V matrix is p-by-p if you asked for #ComputeFullV, and is p-by-m if you asked for ComputeThinV.</p>
<p>The <em>m</em> first columns of <em>V</em> are the right singular vectors of the matrix being decomposed.</p>
<p>This method asserts that you asked for <em>V</em> to be computed. </p>
<p>References <a class="el" href="classEigen_1_1SVDBase.html#a92e99646eefbeb071ef220841555a703">SVDBase< _MatrixType >::computeV()</a>.</p>
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<td class="memname">Index nonzeroSingularValues </td>
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<dl class="section return"><dt>Returns</dt><dd>the number of singular values that are not exactly 0 </dd></dl>
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<td class="memname">const SingularValuesType& singularValues </td>
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<dl class="section return"><dt>Returns</dt><dd>the vector of singular values.</dd></dl>
<p>For the SVD decomposition of a n-by-p matrix, letting <em>m</em> be the minimum of <em>n</em> and <em>p</em>, the returned vector has size <em>m</em>. Singular values are always sorted in decreasing order. </p>
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<td class="memname">const internal::solve_retval<<a class="el" href="classEigen_1_1JacobiSVD.html">JacobiSVD</a>, Rhs> solve </td>
<td>(</td>
<td class="paramtype">const MatrixBase< Rhs > & </td>
<td class="paramname"><em>b</em></td><td>)</td>
<td> const</td>
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<dl class="section return"><dt>Returns</dt><dd>a (least squares) solution of <img class="formulaInl" alt="$ A x = b $" src="form_64.png"/> using the current SVD decomposition of A.</dd></dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">b</td><td>the right-hand-side of the equation to solve.</td></tr>
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<dl class="section note"><dt>Note</dt><dd>Solving requires both U and V to be computed. Thin U and V are enough, there is no need for full U or V.</dd>
<dd>
SVD solving is implicitly least-squares. Thus, this method serves both purposes of exact solving and least-squares solving. In other words, the returned solution is guaranteed to minimize the Euclidean norm <img class="formulaInl" alt="$ \Vert A x - b \Vert $" src="form_92.png"/>. </dd></dl>
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<hr/>The documentation for this class was generated from the following file:<ul>
<li><a class="el" href="JacobiSVD_8h_source.html">JacobiSVD.h</a></li>
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