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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. To get an overview of the true relative speed of the different decomposition, 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 Proven Yes - Yes Excellent -
HouseholderQR - Fast Depends on condition number - Orthogonalization Yes Excellent Blocking
ColPivHouseholderQR - Fast Good Yes Orthogonalization Yes Excellent Soon: blocking
FullPivHouseholderQR - Slow Proven Yes Orthogonalization Yes Average -
LLT Positive definite Very fast Depends on condition number - - Yes Excellent Blocking
LDLT Positive or negative semidefinite1 Very fast Good - - Yes Excellent Soon: blocking
\n Singular values and eigenvalues decompositions
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 Soon: blocking
HessenbergDecomposition Square Average Good - - - Good Soon: blocking
\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 TopicLinAlgTerminology Terminology
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.