 Eigen  3.3.7 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 page . To get an overview of the true relative speed of the different decompositions, check this benchmark .

# 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

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

-

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

Notes:

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

# Terminology

For a real matrix, selfadjoint is a synonym for symmetric. For a complex matrix, selfadjoint is a synonym for hermitian. More generally, a matrix is selfadjoint if and only if it is equal to its adjoint . The adjoint is also called the conjugate transpose.
Positive/negative definite
A selfadjoint matrix is positive definite if for any non zero vector . In the same vein, it is negative definite if for any non zero vector Positive/negative semidefinite

A selfadjoint matrix is positive semi-definite if for any non zero vector . In the same vein, it is negative semi-definite if for any non zero vector 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 algortihm itself is not parallelized, but that it relies on parallelized matrix-matrix product rountines.
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.