<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/> <meta http-equiv="X-UA-Compatible" content="IE=9"/> <meta name="generator" content="Doxygen 1.8.5"/> <title>Eigen: Catalogue of dense decompositions</title> <link href="tabs.css" rel="stylesheet" type="text/css"/> <script type="text/javascript" src="jquery.js"></script> <script type="text/javascript" src="dynsections.js"></script> <link href="navtree.css" rel="stylesheet" type="text/css"/> <script type="text/javascript" src="resize.js"></script> <script type="text/javascript" src="navtree.js"></script> <script type="text/javascript"> $(document).ready(initResizable); $(window).load(resizeHeight); </script> <link href="search/search.css" rel="stylesheet" type="text/css"/> <script type="text/javascript" src="search/search.js"></script> <script type="text/javascript"> $(document).ready(function() { searchBox.OnSelectItem(0); }); </script> <link href="doxygen.css" rel="stylesheet" type="text/css" /> <link href="eigendoxy.css" rel="stylesheet" type="text/css"> <!-- --> <script type="text/javascript" src="eigen_navtree_hacks.js"></script> <!-- <script type="text/javascript"> --> <!-- </script> --> </head> <body> <div id="top"><!-- do not remove this div, it is closed by doxygen! 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">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> </div><!-- contents --> </div><!-- doc-content --> <!-- start footer part --> <div id="nav-path" class="navpath"><!-- id is needed for treeview function! --> <ul> <li class="footer">Generated on Mon Oct 28 2013 11:04:27 for Eigen by <a href="http://www.doxygen.org/index.html"> <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.5 </li> </ul> </div> <!-- Piwik --> <!-- <script type="text/javascript"> var pkBaseURL = (("https:" == document.location.protocol) ? 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