A cross-product free Jacobi-Davidson type method for computing a partial generalized singular value decomposition (GSVD) of a large matrix pair

TL;DR

This paper introduces a novel cross-product free Jacobi-Davidson method for efficiently computing partial GSVD of large matrices without forming cross-product matrices, improving accuracy and computational efficiency.

Contribution

It proposes a new CPF-JD method that avoids forming cross-product matrices, with convergence analysis and a practical algorithm for multiple GSVD components.

Findings

The method effectively computes partial GSVDs for large matrices.

Numerical experiments demonstrate the algorithm's efficiency.

Convergence results support the method's reliability.

Abstract

A Cross-Product Free (CPF) Jacobi-Davidson (JD) type method is proposed to compute a partial generalized singular value decomposition (GSVD) of a large regular matrix pair $(A,B)$. It implicitly solves the mathematically equivalent generalized eigenvalue problem of $(A^TA,B^TB)$ but does not explicitly form the cross-product matrices and thus avoids the possible accuracy loss of the computed generalized singular values and generalized singular vectors. The method is an inner-outer iteration method, where the expansion of the right searching subspace forms the inner iterations that approximately solve the correction equations involved and the outer iterations extract approximate GSVD components with respect to the subspaces. Some convergence results are established for the inner and outer iterations, based on some of which practical stopping criteria are designed for the inner iterations. A thick-restart CPF-JDGSVD algorithm with deflation is developed to compute several GSVD components. Numerical experiments illustrate the efficiency of the algorithm.