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

Jinzhi Huang, Zhongxiao Jia

A Cross-Product Free (CPF) Jacobi-Davidson (JD) type method is proposed to compute a partial generalized singular value decomposition (GSVD) of a large matrix pair $(A,B)$, which is referred to as the CPF-JDGSVD method. By computing the thin QR factorizations of two certain matrices, the method implicitly solves the mathematically equivalent generalized eigenvalue problem of a certain cross-product matrix pair but does not explicitly form the cross-product matrices and thus avoids the accuracy loss of the computed generalized singular values and generalized singular vectors. At each step, the right searching subspace is expanded by approximately solving a correction equation iteratively, called inner iterations, and the two left searching subspaces are constructed by premultiplying the right one by $A$ and $B$, respectively. The extraction steps of approximate GSVD components with respect to given searching subspaces constitute outer iterations. A thick-restart CPF-JDGSVD algorithm with deflation is developed to compute several GSVD components, and some convergence results are established on inner and outer iterations, which are exploited to design practical stopping criteria for inner iterations. Numerical experiments illustrate the effectiveness of the algorithm.

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