A Contour Integral-Based Algorithm for Computing Generalized Singular Values

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 pair. Mathematically, the generalized singular values of a matrix pair are the eigenvalues of an equivalent Hermitian-definite matrix pencil, known as the Jordan-Wielandt matrix pencil. However, direct application of the FEAST algorithm 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.