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<div class="header">
  <div class="headertitle">
<div class="title">Catalogue of dense decompositions<div class="ingroups"><a class="el" href="group__DenseLinearSolvers__chapter.html">Dense linear problems and decompositions</a></div></div>  </div>
</div><!--header-->
<div class="contents">
<p>This page presents a catalogue of the dense matrix decompositions offered by <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a>. For an introduction on linear solvers and decompositions, check this <a class="el" href="group__TutorialLinearAlgebra.html">page </a>.</p>
<h1><a class="anchor" id="TopicLinAlgBigTable"></a>
Catalogue of decompositions offered by Eigen</h1>
<table  class="manual-vl">
<tr>
<th class="meta"></th><th class="meta" colspan="5">Generic information, not Eigen-specific </th><th class="meta" colspan="3"><p class="starttd">Eigen-specific </p>
<p class="endtd"></p>
</th></tr>
<tr>
<th>Decomposition </th><th>Requirements on the matrix </th><th>Speed </th><th>Algorithm reliability and accuracy </th><th>Rank-revealing </th><th>Allows to compute (besides linear solving) </th><th>Linear solver provided by <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a> </th><th>Maturity of <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a>'s implementation </th><th><p class="starttd">Optimizations </p>
<p class="endtd"></p>
</th></tr>
<tr>
<td><a class="el" href="classEigen_1_1PartialPivLU.html" title="LU decomposition of a matrix with partial pivoting, and related features. ">PartialPivLU</a> </td><td>Invertible </td><td>Fast </td><td>Depends on condition number </td><td>- </td><td>- </td><td>Yes </td><td>Excellent </td><td><p class="starttd">Blocking, Implicit MT </p>
<p class="endtd"></p>
</td></tr>
<tr class="alt">
<td><a class="el" href="classEigen_1_1FullPivLU.html" title="LU decomposition of a matrix with complete pivoting, and related features. ">FullPivLU</a> </td><td>- </td><td>Slow </td><td>Proven </td><td>Yes </td><td>- </td><td>Yes </td><td>Excellent </td><td><p class="starttd">- </p>
<p class="endtd"></p>
</td></tr>
<tr>
<td><a class="el" href="classEigen_1_1HouseholderQR.html" title="Householder QR decomposition of a matrix. ">HouseholderQR</a> </td><td>- </td><td>Fast </td><td>Depends on condition number </td><td>- </td><td>Orthogonalization </td><td>Yes </td><td>Excellent </td><td><p class="starttd">Blocking </p>
<p class="endtd"></p>
</td></tr>
<tr class="alt">
<td><a class="el" href="classEigen_1_1ColPivHouseholderQR.html" title="Householder rank-revealing QR decomposition of a matrix with column-pivoting. ">ColPivHouseholderQR</a> </td><td>- </td><td>Fast </td><td>Good </td><td>Yes </td><td>Orthogonalization </td><td>Yes </td><td>Excellent </td><td><p class="starttd"><em>Soon: blocking</em> </p>
<p class="endtd"></p>
</td></tr>
<tr>
<td><a class="el" href="classEigen_1_1FullPivHouseholderQR.html" title="Householder rank-revealing QR decomposition of a matrix with full pivoting. ">FullPivHouseholderQR</a> </td><td>- </td><td>Slow </td><td>Proven </td><td>Yes </td><td>Orthogonalization </td><td>Yes </td><td>Average </td><td><p class="starttd">- </p>
<p class="endtd"></p>
</td></tr>
<tr class="alt">
<td><a class="el" href="classEigen_1_1LLT.html" title="Standard Cholesky decomposition (LL^T) of a matrix and associated features. ">LLT</a> </td><td>Positive definite </td><td>Very fast </td><td>Depends on condition number </td><td>- </td><td>- </td><td>Yes </td><td>Excellent </td><td><p class="starttd">Blocking </p>
<p class="endtd"></p>
</td></tr>
<tr>
<td><a class="el" href="classEigen_1_1LDLT.html" title="Robust Cholesky decomposition of a matrix with pivoting. ">LDLT</a> </td><td>Positive or negative semidefinite<sup><a href="#note1">1</a></sup> </td><td>Very fast </td><td>Good </td><td>- </td><td>- </td><td>Yes </td><td>Excellent </td><td><p class="starttd"><em>Soon: blocking</em> </p>
<p class="endtd"></p>
</td></tr>
<tr>
<th class="inter" colspan="9"><p class="starttd"><br/>
 Singular values and eigenvalues decompositions</p>
<p class="endtd"></p>
</th></tr>
<tr>
<td><a class="el" href="classEigen_1_1JacobiSVD.html" title="Two-sided Jacobi SVD decomposition of a rectangular matrix. ">JacobiSVD</a> (two-sided) </td><td>- </td><td>Slow (but fast for small matrices) </td><td>Excellent-Proven<sup><a href="#note3">3</a></sup> </td><td>Yes </td><td>Singular values/vectors, least squares </td><td>Yes (and does least squares) </td><td>Excellent </td><td><p class="starttd">R-SVD </p>
<p class="endtd"></p>
</td></tr>
<tr class="alt">
<td><a class="el" href="classEigen_1_1SelfAdjointEigenSolver.html" title="Computes eigenvalues and eigenvectors of selfadjoint matrices. ">SelfAdjointEigenSolver</a> </td><td>Self-adjoint </td><td>Fast-average<sup><a href="#note2">2</a></sup> </td><td>Good </td><td>Yes </td><td>Eigenvalues/vectors </td><td>- </td><td>Good </td><td><p class="starttd"><em>Closed forms for 2x2 and 3x3</em> </p>
<p class="endtd"></p>
</td></tr>
<tr>
<td><a class="el" href="classEigen_1_1ComplexEigenSolver.html" title="Computes eigenvalues and eigenvectors of general complex matrices. ">ComplexEigenSolver</a> </td><td>Square </td><td>Slow-very slow<sup><a href="#note2">2</a></sup> </td><td>Depends on condition number </td><td>Yes </td><td>Eigenvalues/vectors </td><td>- </td><td>Average </td><td><p class="starttd">- </p>
<p class="endtd"></p>
</td></tr>
<tr class="alt">
<td><a class="el" href="classEigen_1_1EigenSolver.html" title="Computes eigenvalues and eigenvectors of general matrices. ">EigenSolver</a> </td><td>Square and real </td><td>Average-slow<sup><a href="#note2">2</a></sup> </td><td>Depends on condition number </td><td>Yes </td><td>Eigenvalues/vectors </td><td>- </td><td>Average </td><td><p class="starttd">- </p>
<p class="endtd"></p>
</td></tr>
<tr>
<td><a class="el" href="classEigen_1_1GeneralizedSelfAdjointEigenSolver.html" title="Computes eigenvalues and eigenvectors of the generalized selfadjoint eigen problem. ">GeneralizedSelfAdjointEigenSolver</a> </td><td>Square </td><td>Fast-average<sup><a href="#note2">2</a></sup> </td><td>Depends on condition number </td><td>- </td><td>Generalized eigenvalues/vectors </td><td>- </td><td>Good </td><td><p class="starttd">- </p>
<p class="endtd"></p>
</td></tr>
<tr>
<th class="inter" colspan="9"><p class="starttd"><br/>
 Helper decompositions</p>
<p class="endtd"></p>
</th></tr>
<tr>
<td><a class="el" href="classEigen_1_1RealSchur.html" title="Performs a real Schur decomposition of a square matrix. ">RealSchur</a> </td><td>Square and real </td><td>Average-slow<sup><a href="#note2">2</a></sup> </td><td>Depends on condition number </td><td>Yes </td><td>- </td><td>- </td><td>Average </td><td><p class="starttd">- </p>
<p class="endtd"></p>
</td></tr>
<tr class="alt">
<td><a class="el" href="classEigen_1_1ComplexSchur.html" title="Performs a complex Schur decomposition of a real or complex square matrix. ">ComplexSchur</a> </td><td>Square </td><td>Slow-very slow<sup><a href="#note2">2</a></sup> </td><td>Depends on condition number </td><td>Yes </td><td>- </td><td>- </td><td>Average </td><td><p class="starttd">- </p>
<p class="endtd"></p>
</td></tr>
<tr class="alt">
<td><a class="el" href="classEigen_1_1Tridiagonalization.html" title="Tridiagonal decomposition of a selfadjoint matrix. ">Tridiagonalization</a> </td><td>Self-adjoint </td><td>Fast </td><td>Good </td><td>- </td><td>- </td><td>- </td><td>Good </td><td><p class="starttd"><em>Soon: blocking</em> </p>
<p class="endtd"></p>
</td></tr>
<tr>
<td><a class="el" href="classEigen_1_1HessenbergDecomposition.html" title="Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation. ">HessenbergDecomposition</a> </td><td>Square </td><td>Average </td><td>Good </td><td>- </td><td>- </td><td>- </td><td>Good </td><td><p class="starttd"><em>Soon: blocking</em> </p>
<p class="endtd"></p>
</td></tr>
</table>
<p><b>Notes:</b> </p>
<ul>
<li>
<a class="anchor" id="note1"></a><b>1</b>: There exist two variants of the <a class="el" href="classEigen_1_1LDLT.html" title="Robust Cholesky decomposition of a matrix with pivoting. ">LDLT</a> algorithm. <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a>'s one produces a pure diagonal D matrix, and therefore it cannot handle indefinite matrices, unlike Lapack's one which produces a block diagonal D matrix. </li>
<li>
<a class="anchor" id="note2"></a><b>2</b>: Eigenvalues, SVD and Schur decompositions rely on iterative algorithms. Their convergence speed depends on how well the eigenvalues are separated. </li>
<li>
<a class="anchor" id="note3"></a><b>3</b>: Our <a class="el" href="classEigen_1_1JacobiSVD.html" title="Two-sided Jacobi SVD decomposition of a rectangular matrix. ">JacobiSVD</a> is two-sided, making for proven and optimal precision for square matrices. For non-square matrices, we have to use a QR preconditioner first. The default choice, <a class="el" href="classEigen_1_1ColPivHouseholderQR.html" title="Householder rank-revealing QR decomposition of a matrix with column-pivoting. ">ColPivHouseholderQR</a>, is already very reliable, but if you want it to be proven, use <a class="el" href="classEigen_1_1FullPivHouseholderQR.html" title="Householder rank-revealing QR decomposition of a matrix with full pivoting. ">FullPivHouseholderQR</a> instead. </li>
</ul>
<h1><a class="anchor" id="TopicLinAlgTerminology"></a>
Terminology</h1>
<dl>
<dt><b>Selfadjoint</b> </dt>
<dd>For a real matrix, selfadjoint is a synonym for symmetric. For a complex matrix, selfadjoint is a synonym for <em>hermitian</em>. More generally, a matrix <img class="formulaInl" alt="$ A $" src="form_1.png"/> is selfadjoint if and only if it is equal to its adjoint <img class="formulaInl" alt="$ A^* $" src="form_177.png"/>. The adjoint is also called the <em>conjugate</em> <em>transpose</em>.  </dd>
<dt><b>Positive/negative definite</b> </dt>
<dd>A selfadjoint matrix <img class="formulaInl" alt="$ A $" src="form_1.png"/> is positive definite if <img class="formulaInl" alt="$ v^* A v > 0 $" src="form_178.png"/> for any non zero vector <img class="formulaInl" alt="$ v $" src="form_13.png"/>. In the same vein, it is negative definite if <img class="formulaInl" alt="$ v^* A v < 0 $" src="form_179.png"/> for any non zero vector <img class="formulaInl" alt="$ v $" src="form_13.png"/>  </dd>
<dt><b>Positive/negative semidefinite</b> </dt>
<dd><p class="startdd">A selfadjoint matrix <img class="formulaInl" alt="$ A $" src="form_1.png"/> is positive semi-definite if <img class="formulaInl" alt="$ v^* A v \ge 0 $" src="form_180.png"/> for any non zero vector <img class="formulaInl" alt="$ v $" src="form_13.png"/>. In the same vein, it is negative semi-definite if <img class="formulaInl" alt="$ v^* A v \le 0 $" src="form_181.png"/> for any non zero vector <img class="formulaInl" alt="$ v $" src="form_13.png"/> </p>
<p class="enddd"></p>
</dd>
<dt><b>Blocking</b> </dt>
<dd>Means the algorithm can work per block, whence guaranteeing a good scaling of the performance for large matrices. </dd>
<dt><b>Implicit Multi Threading (MT)</b> </dt>
<dd>Means the algorithm can take advantage of multicore processors via OpenMP. "Implicit" means the algortihm itself is not parallelized, but that it relies on parallelized matrix-matrix product rountines. </dd>
<dt><b>Explicit Multi Threading (MT)</b> </dt>
<dd>Means the algorithm is explicitely parallelized to take advantage of multicore processors via OpenMP. </dd>
<dt><b>Meta-unroller</b> </dt>
<dd>Means the algorithm is automatically and explicitly unrolled for very small fixed size matrices. </dd>
<dt><b></b> </dt>
<dd></dd>
</dl>
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