Two harmonic Jacobi--Davidson methods for computing a partial generalized singular value decomposition of a large matrix pair

TL;DR

This paper introduces two new harmonic Jacobi--Davidson algorithms, CPF-HJDGSVD and IF-HJDGSVD, for efficiently computing partial generalized singular value decompositions of large matrix pairs, especially for interior singular values.

Contribution

The paper proposes two novel harmonic Jacobi--Davidson algorithms that improve convergence and applicability for computing interior GSVD components of large matrices.

Findings

Harmonic JDGSVD algorithms converge more regularly than standard methods.

Numerical experiments show the superiority of the proposed algorithms.

Algorithms effectively compute multiple GSVD components with deflation techniques.

Abstract

Two harmonic extraction based Jacobi--Davidson (JD) type algorithms are proposed to compute a partial generalized singular value decomposition (GSVD) of a large regular matrix pair. They are called cross product-free (CPF) and inverse-free (IF) harmonic JDGSVD algorithms, abbreviated as CPF-HJDGSVD and IF-HJDGSVD, respectively. Compared with the standard extraction based JDGSVD algorithm, the harmonic extraction based algorithms converge more regularly and suit better for computing GSVD components corresponding to interior generalized singular values. Thick-restart CPF-HJDGSVD and IF-HJDGSVD algorithms with some deflation and purgation techniques are developed to compute more than one GSVD components. Numerical experiments confirm the superiority of CPF-HJDGSVD and IF-HJDGSVD to the standard extraction based JDGSVD algorithm.