A Preconditioned Hybrid SVD Method for Computing Accurately Singular Triplets of Large Matrices

TL;DR

The paper introduces PHSVDS, a hybrid preconditioned SVD method that efficiently computes accurate singular triplets of large sparse matrices by combining two stages for improved convergence and precision.

Contribution

It presents a novel two-stage hybrid approach, PHSVDS, that leverages preconditioning and eigensolvers to accurately compute both largest and smallest singular triplets.

Findings

PHSVDS achieves high accuracy for singular triplets of large matrices.

The method demonstrates improved convergence speed over existing techniques.

Numerical experiments confirm robustness and efficiency.

Abstract

The computation of a few singular triplets of large, sparse matrices is a challenging task, especially when the smallest magnitude singular values are needed in high accuracy. Most recent efforts try to address this problem through variations of the Lanczos bidiagonalization method, but they are still challenged even for medium matrix sizes due to the difficulty of the problem. We propose a novel SVD approach that can take advantage of preconditioning and of any well designed eigensolver to compute both largest and smallest singular triplets. Accuracy and efficiency is achieved through a hybrid, two-stage meta-method, PHSVDS. In the first stage, PHSVDS solves the normal equations up to the best achievable accuracy. If further accuracy is required, the method switches automatically to an eigenvalue problem with the augmented matrix. Thus it combines the advantages of the two stages, faster convergence and accuracy, respectively. For the augmented matrix, solving the interior eigenvalue is facilitated by a proper use of the good initial guesses from the first stage and an efficient implementation of the refined projection method. We also discuss how to precondition PHSVDS and to cope with some issues that arise. Numerical experiments illustrate the efficiency and robustness of the method.