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A Contour Integral-Based Algorithm for Computing Generalized Singular Values

Yuqi Liu, Xinyu Shan, Meiyue Shao

TL;DR

This paper develops contour-integral FEAST-based algorithms to compute a few singular values or generalized singular values via the Jordan--Wielandt pencil. It analyzes projection strategies, introduces a robust augmented contour scheme, and extends FEAST to GSVD with Rayleigh--Ritz projections. Numerical experiments on sparse matrices show rapid convergence (typically 2–4 iterations) and high accuracy, outperforming Jacobi--Davidson in the tested cases. The work enables efficient partial SVD/GSVD computation and suggests avenues for spectral slicing and mixed-precision implementations in large-scale settings.

Abstract

We propose a contour integral-based algorithm for computing a few singular values of a matrix or a few generalized singular values of a matrix pencil. Mathematically, the generalized singular values of a matrix pencil are the eigenvalues of an equivalent Hermitian-definite matrix pencil, known as the Jordan-Wielandt matrix pencil. However, direct application of the FEAST solver does not fully exploit the structure of this problem. We analyze several projection strategies on the Jordan-Wielandt matrix pencil, and propose an effective and robust scheme tailored to GSVD. Both theoretical analysis and numerical experiments demonstrate that our algorithm achieves rapid convergence and satisfactory accuracy.

A Contour Integral-Based Algorithm for Computing Generalized Singular Values

TL;DR

This paper develops contour-integral FEAST-based algorithms to compute a few singular values or generalized singular values via the Jordan--Wielandt pencil. It analyzes projection strategies, introduces a robust augmented contour scheme, and extends FEAST to GSVD with Rayleigh--Ritz projections. Numerical experiments on sparse matrices show rapid convergence (typically 2–4 iterations) and high accuracy, outperforming Jacobi--Davidson in the tested cases. The work enables efficient partial SVD/GSVD computation and suggests avenues for spectral slicing and mixed-precision implementations in large-scale settings.

Abstract

We propose a contour integral-based algorithm for computing a few singular values of a matrix or a few generalized singular values of a matrix pencil. Mathematically, the generalized singular values of a matrix pencil are the eigenvalues of an equivalent Hermitian-definite matrix pencil, known as the Jordan-Wielandt matrix pencil. However, direct application of the FEAST solver does not fully exploit the structure of this problem. We analyze several projection strategies on the Jordan-Wielandt matrix pencil, and propose an effective and robust scheme tailored to GSVD. Both theoretical analysis and numerical experiments demonstrate that our algorithm achieves rapid convergence and satisfactory accuracy.
Paper Structure (23 sections, 4 theorems, 125 equations, 5 figures, 3 tables, 2 algorithms)

This paper contains 23 sections, 4 theorems, 125 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized singular value decomposition
  4. The FEAST algorithm
  5. A FEAST-SVD Algorithm
  6. Structured Galerkin condition and Rayleigh--Ritz projection
  7. Choice of the spectral projector
  8. A simple spectral projector
  9. A simple spectral projector with Rayleigh--Ritz projection
  10. A pair of contours
  11. An augmented scheme with a pair of contours
  12. Trace estimation
  13. A FEAST-GSVD Algorithm
  14. Rayleigh--Ritz projection
  15. A FEAST algorithm for GSVD
  16. ...and 8 more sections

Key Result

Lemma 1

Let $M\in\mathbb C^{m\times m}$ be Hermitian and positive semidefinite. Then

Figures (5)

  • Figure 1: An ellipse contour with eight quadrature nodes (i.e., $N=8$).
  • Figure 2: Convergence history for SVD experiments.
  • Figure 3: Convergence history for GSVD experiments.
  • Figure 4: Comparison on four spectral projectors applied to compute the GSVD of GL7d12: (a) using random initial guess; (b) using artificial initial guess \ref{['eq:artificial1']}. (c) using artificial initial guess \ref{['eq:artificial2']}.
  • Figure 5: Compute the generalized singular values of plat1919 in the interval $(10^{-4},10^{-3})$ with low-precision initial guesses by Algorithm \ref{['alg:FEAST-GSVD']}: (a) using artificial initial guesses \ref{['eq:artificial3']}; (b) using MATLAB's ${\tt{eigs}}(A^* A, B^* B)$ as the initial guess.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof