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| author | Stanislaw Halik <sthalik@misaki.pl> | 2019-03-03 21:09:10 +0100 |
|---|---|---|
| committer | Stanislaw Halik <sthalik@misaki.pl> | 2019-03-03 21:10:13 +0100 |
| commit | f0238cfb6997c4acfc2bd200de7295f3fa36968f (patch) | |
| tree | b215183760e4f615b9c1dabc1f116383b72a1b55 /eigen/doc/TopicLinearAlgebraDecompositions.dox | |
| parent | 543edd372a5193d04b3de9f23c176ab439e51b31 (diff) | |
don't index Eigen
Diffstat (limited to 'eigen/doc/TopicLinearAlgebraDecompositions.dox')
| -rw-r--r-- | eigen/doc/TopicLinearAlgebraDecompositions.dox | 275 |
1 files changed, 0 insertions, 275 deletions
diff --git a/eigen/doc/TopicLinearAlgebraDecompositions.dox b/eigen/doc/TopicLinearAlgebraDecompositions.dox deleted file mode 100644 index d9db677..0000000 --- a/eigen/doc/TopicLinearAlgebraDecompositions.dox +++ /dev/null @@ -1,275 +0,0 @@ -namespace Eigen { - -/** \eigenManualPage TopicLinearAlgebraDecompositions Catalogue of dense decompositions - -This page presents a catalogue of the dense matrix decompositions offered by Eigen. -For an introduction on linear solvers and decompositions, check this \link TutorialLinearAlgebra page \endlink. -To get an overview of the true relative speed of the different decompositions, check this \link DenseDecompositionBenchmark benchmark \endlink. - -\section TopicLinAlgBigTable Catalogue of decompositions offered by Eigen - -<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">Eigen-specific</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 Eigen</th> - <th>Maturity of Eigen's implementation</th> - <th>Optimizations</th> - </tr> - - <tr> - <td>PartialPivLU</td> - <td>Invertible</td> - <td>Fast</td> - <td>Depends on condition number</td> - <td>-</td> - <td>-</td> - <td>Yes</td> - <td>Excellent</td> - <td>Blocking, Implicit MT</td> - </tr> - - <tr class="alt"> - <td>FullPivLU</td> - <td>-</td> - <td>Slow</td> - <td>Proven</td> - <td>Yes</td> - <td>-</td> - <td>Yes</td> - <td>Excellent</td> - <td>-</td> - </tr> - - <tr> - <td>HouseholderQR</td> - <td>-</td> - <td>Fast</td> - <td>Depends on condition number</td> - <td>-</td> - <td>Orthogonalization</td> - <td>Yes</td> - <td>Excellent</td> - <td>Blocking</td> - </tr> - - <tr class="alt"> - <td>ColPivHouseholderQR</td> - <td>-</td> - <td>Fast</td> - <td>Good</td> - <td>Yes</td> - <td>Orthogonalization</td> - <td>Yes</td> - <td>Excellent</td> - <td><em>Soon: blocking</em></td> - </tr> - - <tr> - <td>FullPivHouseholderQR</td> - <td>-</td> - <td>Slow</td> - <td>Proven</td> - <td>Yes</td> - <td>Orthogonalization</td> - <td>Yes</td> - <td>Average</td> - <td>-</td> - </tr> - - <tr class="alt"> - <td>LLT</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>Blocking</td> - </tr> - - <tr> - <td>LDLT</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><em>Soon: blocking</em></td> - </tr> - - <tr><th class="inter" colspan="9">\n Singular values and eigenvalues decompositions</th></tr> - - <tr> - <td>BDCSVD (divide \& conquer)</td> - <td>-</td> - <td>One of the fastest SVD algorithms</td> - <td>Excellent</td> - <td>Yes</td> - <td>Singular values/vectors, least squares</td> - <td>Yes (and does least squares)</td> - <td>Excellent</td> - <td>Blocked bidiagonalization</td> - </tr> - - <tr> - <td>JacobiSVD (two-sided)</td> - <td>-</td> - <td>Slow (but fast for small matrices)</td> - <td>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>R-SVD</td> - </tr> - - <tr class="alt"> - <td>SelfAdjointEigenSolver</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>Excellent</td> - <td><em>Closed forms for 2x2 and 3x3</em></td> - </tr> - - <tr> - <td>ComplexEigenSolver</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>-</td> - </tr> - - <tr class="alt"> - <td>EigenSolver</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>-</td> - </tr> - - <tr> - <td>GeneralizedSelfAdjointEigenSolver</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>-</td> - </tr> - - <tr><th class="inter" colspan="9">\n Helper decompositions</th></tr> - - <tr> - <td>RealSchur</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>-</td> - </tr> - - <tr class="alt"> - <td>ComplexSchur</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>-</td> - </tr> - - <tr class="alt"> - <td>Tridiagonalization</td> - <td>Self-adjoint</td> - <td>Fast</td> - <td>Good</td> - <td>-</td> - <td>-</td> - <td>-</td> - <td>Good</td> - <td><em>Soon: blocking</em></td> - </tr> - - <tr> - <td>HessenbergDecomposition</td> - <td>Square</td> - <td>Average</td> - <td>Good</td> - <td>-</td> - <td>-</td> - <td>-</td> - <td>Good</td> - <td><em>Soon: blocking</em></td> - </tr> - -</table> - -\b Notes: -<ul> -<li><a name="note1">\b 1: </a>There exist two variants of the LDLT algorithm. Eigen'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 name="note2">\b 2: </a>Eigenvalues, SVD and Schur decompositions rely on iterative algorithms. Their convergence speed depends on how well the eigenvalues are separated.</li> -<li><a name="note3">\b 3: </a>Our JacobiSVD 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, ColPivHouseholderQR, is already very reliable, but if you want it to be proven, use FullPivHouseholderQR instead. -</ul> - -\section TopicLinAlgTerminology Terminology - -<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. - More generally, a matrix \f$ A \f$ is selfadjoint if and only if it is equal to its adjoint \f$ A^* \f$. The adjoint is also called the \em conjugate \em transpose. </dd> - <dt><b>Positive/negative definite</b></dt> - <dd>A selfadjoint matrix \f$ A \f$ is positive definite if \f$ v^* A v > 0 \f$ for any non zero vector \f$ v \f$. - In the same vein, it is negative definite if \f$ v^* A v < 0 \f$ for any non zero vector \f$ v \f$ </dd> - <dt><b>Positive/negative semidefinite</b></dt> - <dd>A selfadjoint matrix \f$ A \f$ is positive semi-definite if \f$ v^* A v \ge 0 \f$ for any non zero vector \f$ v \f$. - In the same vein, it is negative semi-definite if \f$ v^* A v \le 0 \f$ for any non zero vector \f$ v \f$ </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 explicitly 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> - - -*/ - -} |
