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

	Generic information, not Eigen-specific					Eigen-specific
Decomposition	Requirements on the matrix	Speed	Algorithm reliability and accuracy	Rank-revealing	Allows to compute (besides linear solving)	Linear solver provided by Eigen	Maturity of Eigen's implementation	Optimizations
PartialPivLU	Invertible	Fast	Depends on condition number	-	-	Yes	Excellent	Blocking, Implicit MT
FullPivLU	-	Slow (no blocking)	Proven	Yes	Rank, kernel, image	Yes	Excellent	-
HouseholderQR	-	Fast	Depends on condition number	-	Orthogonalization	Yes	Excellent	Blocking
ColPivHouseholderQR	-	Fast	Good	Yes	Orthogonalization	Yes	Excellent	-
FullPivHouseholderQR	-	Slow (no blocking)	Proven	Yes	Orthogonalization	Yes	Average	-
CompleteOrthogonalDecomposition	-	Fast	Good	Yes	Orthogonalization	Yes	Excellent	-
LLT	Positive definite	Very fast	Depends on condition number	-	-	Yes	Excellent	Blocking
LDLT	Positive or negative semidefinite¹	Very fast	Good	-	-	Yes	Excellent	-
\n Singular values and eigenvalues decompositions
BDCSVD (divide \& conquer)	-	One of the fastest SVD algorithms	Excellent	Yes	Singular values/vectors, least squares	Yes (and does least squares)	Excellent	Blocked bidiagonalization
JacobiSVD (two-sided)	-	Slow (but fast for small matrices)	Proven³	Yes	Singular values/vectors, least squares	Yes (and does least squares)	Excellent	R-SVD
SelfAdjointEigenSolver	Self-adjoint	Fast-average²	Good	Yes	Eigenvalues/vectors	-	Excellent	Closed forms for 2x2 and 3x3
ComplexEigenSolver	Square	Slow-very slow²	Depends on condition number	Yes	Eigenvalues/vectors	-	Average	-
EigenSolver	Square and real	Average-slow²	Depends on condition number	Yes	Eigenvalues/vectors	-	Average	-
GeneralizedSelfAdjointEigenSolver	Square	Fast-average²	Depends on condition number	-	Generalized eigenvalues/vectors	-	Good	-
\n Helper decompositions
RealSchur	Square and real	Average-slow²	Depends on condition number	Yes	-	-	Average	-
ComplexSchur	Square	Slow-very slow²	Depends on condition number	Yes	-	-	Average	-
Tridiagonalization	Self-adjoint	Fast	Good	-	-	-	Good	-
HessenbergDecomposition	Square	Average	Good	-	-	-	Good	-

\b Notes:

\b 1: 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.
\b 2: Eigenvalues, SVD and Schur decompositions rely on iterative algorithms. Their convergence speed depends on how well the eigenvalues are separated.
\b 3: 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.

\section TopicLinAlgPracticalGuidance Practical guidance The following recommendations apply to the most common use cases: \li Symmetric positive definite systems: Use \b LLT. It is the fastest solver and has excellent numerical properties for this class of problems. For semidefinite or nearly singular symmetric systems, use \b LDLT. \li General invertible systems: Use \b PartialPivLU. It uses cache-friendly blocking and implicit multi-threading, making it the fastest general-purpose solver. Partial pivoting is sufficient for virtually all practical problems. \li Least squares (over- or under-determined systems): Use \b CompleteOrthogonalDecomposition as the default. Like the SVD, it robustly computes the minimum-norm solution for rank-deficient and under-determined problems, but at QR-like speed. Use \b BDCSVD when you also need singular values or vectors, not just the least squares solution. \li Full-rank least squares (overdetermined systems): When the matrix is known to be full rank, \b HouseholderQR is the fastest option. For very tall and skinny well-conditioned matrices, solving via the normal equations with \b LLT can be faster still. \li FullPivLU and FullPivHouseholderQR use complete pivoting, which prevents the use of cache-friendly blocking algorithms and makes them significantly slower than their partial/column pivoting counterparts. In practice, complete pivoting rarely provides meaningful accuracy benefits. These decompositions are primarily useful for debugging, pedagogy, or the very rare case where column pivoting is insufficient. \section TopicLinAlgTerminology Terminology

Selfadjoint: 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$. The adjoint is also called the \em conjugate \em transpose.
Positive/negative definite: A selfadjoint matrix \f$ 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$
Positive/negative semidefinite: 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$
Blocking: Means the algorithm can work per block, whence guaranteeing a good scaling of the performance for large matrices.
Implicit Multi Threading (MT): Means the algorithm can take advantage of multicore processors via OpenMP. "Implicit" means the algorithm itself is not parallelized, but that it relies on parallelized matrix-matrix product routines.
Explicit Multi Threading (MT): Means the algorithm is explicitly parallelized to take advantage of multicore processors via OpenMP.
Meta-unroller: Means the algorithm is automatically and explicitly unrolled for very small fixed size matrices.

*/ }