The joint bidiagonalization method for large GSVD computations in finite precision

TL;DR

This paper introduces a joint bidiagonalization method for efficiently computing large generalized singular value decompositions, analyzing its numerical stability, convergence, and residual bounds.

Contribution

It proposes three approaches for approximate GSVD computation, analyzes their numerical properties, and establishes residual bounds and convergence conditions.

Findings

Semiorthogonality suffices for accurate GSVD approximation.

Established a sharp residual norm bound for approximate GSVD vectors.

Demonstrated the effectiveness of the proposed methods through numerical analysis.

Abstract

The joint bidiagonalization (JBD) method has been used to compute some extreme generalized singular values and vectors of a large regular matrix pair $\{A,L\}$, where we propose three approaches to compute approximate generalized singular values and vectors. We make a numerical analysis of the underlying JBD process and establish relationships between it and two mathematically equivalent Lanczos bidiagonalizations in finite precision. Based on the results of numerical analysis, we investigate the convergence of the approximate generalized singular values and vectors of $\{A,L\}$. The results show that, under some mild conditions, the semiorthogonality of Lanczos type vectors suffices to deliver approximate generalized singular values with the same accuracy as the full orthogonality does, meaning that it is only necessary to seek for efficient semiorthogonalization strategies for the JBD process. We also establish a sharp bound for the residual norm of an approximate generalized singular value and corresponding approximate right generalized singular vectors, which can reliably estimate the residual norm without explicitly computing the approximate right generalized singular vectors before the convergence occurs.