Refined and refined harmonic Jacobi--Davidson methods for computing several GSVD components of a large regular matrix pair

TL;DR

This paper introduces three refined Jacobi--Davidson methods with advanced algorithms to efficiently compute multiple GSVD components of large regular matrix pairs, improving stability and convergence.

Contribution

The authors propose three novel refined JDGSVD algorithms with thick-restart, deflation, and purgation, outperforming previous methods in efficiency and convergence stability.

Findings

RCPF-JDGSVD excels at computing extreme GSVD components.

RCPF-HJDGSVD and RIF-HJDGSVD are better suited for interior GSVD components.

Numerical experiments confirm improved efficiency and convergence stability.

Abstract

Three refined and refined harmonic extraction-based Jacobi--Davidson (JD) type methods are proposed, and their thick-restart algorithms with deflation and purgation are developed to compute several generalized singular value decomposition (GSVD) components of a large regular matrix pair. The new methods are called refined cross product-free (RCPF), refined cross product-free harmonic (RCPF-harmonic) and refined inverse-free harmonic (RIF-harmonic) JDGSVD algorithms, abbreviated as RCPF-JDGSVD, RCPF-HJDGSVD and RIF-HJDGSVD, respectively. The new JDGSVD methods are more efficient than the corresponding standard and harmonic extraction-based JDSVD methods proposed previously by the authors, and can overcome the erratic behavior and intrinsic possible non-convergence of the latter ones. Numerical experiments illustrate that RCPF-JDGSVD performs better for the computation of extreme GSVD components while RCPF-HJDGSVD and RIF-HJDGSVD suit better for that of interior GSVD components.