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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.
\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>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>
*/
}
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