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

Jinzhi Huang, Zhongxiao Jia

TL;DR

This work addresses computing multiple GSVD components of a large regular matrix pair $(A,B)$ by introducing three refined Jacobi–Davidson–type methods: RCPF-JDGSVD, RCPF-HJDGSVD, and RIF-HJDGSVD. The authors fuse refined extraction with cross product-free and harmonic extraction strategies and embed them in thick-restart schemes with deflation and purgation to robustly extract several components near a target $\tau$, handling both extreme and interior GSVD components. Empirical results on large sparse matrix pairs show that RCPF-JDGSVD excels for extreme components, while RCPF-HJDGSVD and RIF-HJDGSVD offer superior performance and reliability for interior components, demonstrating improved convergence and efficiency over prior CPF/IF-HJDGSVD methods. The approach thus provides a practical, scalable framework for reliable partial GSVD computations in large-scale applications, with future work focusing on specialized solvers and preconditioners for the correction equations.

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.

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

TL;DR

This work addresses computing multiple GSVD components of a large regular matrix pair by introducing three refined Jacobi–Davidson–type methods: RCPF-JDGSVD, RCPF-HJDGSVD, and RIF-HJDGSVD. The authors fuse refined extraction with cross product-free and harmonic extraction strategies and embed them in thick-restart schemes with deflation and purgation to robustly extract several components near a target , handling both extreme and interior GSVD components. Empirical results on large sparse matrix pairs show that RCPF-JDGSVD excels for extreme components, while RCPF-HJDGSVD and RIF-HJDGSVD offer superior performance and reliability for interior components, demonstrating improved convergence and efficiency over prior CPF/IF-HJDGSVD methods. The approach thus provides a practical, scalable framework for reliable partial GSVD computations in large-scale applications, with future work focusing on specialized solvers and preconditioners for the correction equations.

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.
Paper Structure (13 sections, 1 theorem, 57 equations, 4 figures, 5 tables, 1 algorithm)

This paper contains 13 sections, 1 theorem, 57 equations, 4 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The standard and two harmonic extraction-based JDGSVD methods
  3. The construction and expansion of searching subspaces
  4. The standard extraction approach
  5. The CPF-harmonic extraction approach
  6. The IF-harmonic extraction approach
  7. The refined JDGSVD type methods
  8. Thick-restart refined JDGSVD algorithms with deflation and purgation
  9. Thick-restart
  10. Deflation and purgation
  11. Thick restart refined and refined harmonic JDGSVD algorithms
  12. Numerical experiments
  13. Conclusions

Key Result

Theorem 4.1

Suppose that the current right subspace $\mathcal{X}$ is orthogonal to $\mathcal{R}(Y_c)$. Then the expanded $\mathcal{X}$'s are also orthogonal to $\mathcal{R}(Y_c)$ at subsequent expansion steps.

Figures (4)

  • Figure 5.1: Computing one GSVD component of $(A,B)=(\mathrm{dano3mip}^T,T)$ with $\tau=3.75e+2$.
  • Figure 5.2: Computing one GSVD component of $(A,B)=(\mathrm{plddb}^T,T)$ with $\tau=10$.
  • Figure 5.3: Computing one GSVD component of $(A,B)=(\mathrm{large}^T,T)$ with $\tau=14$.
  • Figure 5.4: Computing one GSVD component of $(A,B)=(\mathrm{nemeth01},D)$ with $\tau=6.5$.

Theorems & Definitions (2)

  • Theorem 4.1
  • proof