A Cross-Product Free Jacobi–Davidson Type Method for Computing a Partial Generalized Singular Value Decomposition of a Large Matrix Pair

Abstract

A cross-product free (CPF) Jacobi–Davidson type method is proposed to compute a partial generalized singular value decomposition (GSVD) of a large regular matrix pair $\{A,B\}$, called CPF-JDGSVD. It implicitly solves the mathematically equivalent generalized eigenvalue problem of the cross-product matrix pair $\{A^TA,B^TB\}$ using the Rayleigh–Ritz projection method but does not form the cross-product matrices explicitly, 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. A convergence result is established for the outer iterations, compact bounds are derived for the condition numbers of the correction equations, and the least solution accuracy requirements on the inner iterations are found, which can maximize the overall efficiency of CPF-JDGSVD as much as possible. Based on them, practical stopping criteria are designed for the inner iterations. A thick-restart CPF-JDGSVD algorithm with deflation and purgation is developed to compute several GSVD components of $\{A,B\}$ associated with the generalized singular values closest to a given target $\tau$. Numerical experiments illustrate the efficiency of the algorithm.