Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The joint bidiagonalization of a matrix pair with inaccurate inner iterations

Haibo Li

TL;DR

This work addresses the impact of inaccurate inner iterations on the joint bidiagonalization (JBD) of a matrix pair $\\\\{A,L\\\}$ used to compute a partial generalized SVD (GSVD). It develops a reorthogonalized JBD (rJBD) method to maintain orthogonality of Lanczos vectors, and provides an error analysis that links inner-solve accuracy $\\tau$ and the conditioning of $C=(A^T,L^T)^T$ via a perturbation bound of $O(\\kappa(C)\\tau)$ on the outer iterations. The results show that GSVD components converge regularly, with their accuracy governed by $\\kappa(C)\\tau$ and spectral gaps, while convergence rate remains robust. The paper also supplies practical guidance on selecting inner-solver tolerances to achieve a desired GSVD accuracy, supported by numerical experiments that corroborate the theory.

Abstract

The joint bidiagonalization (JBD) process iteratively reduces a matrix pair $\{A,L\}$ to two bidiagonal forms simultaneously, which can be used for computing a partial generalized singular value decomposition (GSVD) of $\{A,L\}$. The process has a nested inner-outer iteration structure, where the inner iteration usually can not be computed exactly. In this paper, we study the inaccurately computed inner iterations of JBD by first investigating influence of computational error of the inner iteration on the outer iteration, and then proposing a reorthogonalized JBD (rJBD) process to keep orthogonality of a part of Lanczos vectors. An error analysis of the rJBD is carried out to build up connections with Lanczos bidiagonalizations. The results are then used to investigate convergence and accuracy of the rJBD based GSVD computation. It is shown that the accuracy of computed GSVD components depend on the computing accuracy of inner iterations and condition number of $(A^T,L^T)^T$ while the convergence rate is not affected very much. For practical JBD based GSVD computations, our results can provide a guideline for choosing a proper computing accuracy of inner iterations in order to obtain approximate GSVD components with a desired accuracy. Numerical experiments are made to confirm our theoretical results.

The joint bidiagonalization of a matrix pair with inaccurate inner iterations

TL;DR

This work addresses the impact of inaccurate inner iterations on the joint bidiagonalization (JBD) of a matrix pair used to compute a partial generalized SVD (GSVD). It develops a reorthogonalized JBD (rJBD) method to maintain orthogonality of Lanczos vectors, and provides an error analysis that links inner-solve accuracy and the conditioning of via a perturbation bound of on the outer iterations. The results show that GSVD components converge regularly, with their accuracy governed by and spectral gaps, while convergence rate remains robust. The paper also supplies practical guidance on selecting inner-solver tolerances to achieve a desired GSVD accuracy, supported by numerical experiments that corroborate the theory.

Abstract

The joint bidiagonalization (JBD) process iteratively reduces a matrix pair to two bidiagonal forms simultaneously, which can be used for computing a partial generalized singular value decomposition (GSVD) of . The process has a nested inner-outer iteration structure, where the inner iteration usually can not be computed exactly. In this paper, we study the inaccurately computed inner iterations of JBD by first investigating influence of computational error of the inner iteration on the outer iteration, and then proposing a reorthogonalized JBD (rJBD) process to keep orthogonality of a part of Lanczos vectors. An error analysis of the rJBD is carried out to build up connections with Lanczos bidiagonalizations. The results are then used to investigate convergence and accuracy of the rJBD based GSVD computation. It is shown that the accuracy of computed GSVD components depend on the computing accuracy of inner iterations and condition number of while the convergence rate is not affected very much. For practical JBD based GSVD computations, our results can provide a guideline for choosing a proper computing accuracy of inner iterations in order to obtain approximate GSVD components with a desired accuracy. Numerical experiments are made to confirm our theoretical results.
Paper Structure (2 sections, 1 theorem, 10 equations)

This paper contains 2 sections, 1 theorem, 10 equations.

Table of Contents

  1. Introduction
  2. Joint bidiagonalization and GSVD of a matrix pair

Key Result

Theorem 2.1

\newlabelthm_cs0 Suppose $\mathrm{Rank}(C)=r$. For the column orthonormal matrix $Q$, the CS decomposition of $\{Q_{A}, Q_{L} \}$ is where

Theorems & Definitions (1)

  • Theorem 2.1: CS decomposition