An Implicitly Restarted Joint Bidiagonalization Algorithm for Large GSVD Computations

Abstract

The joint bidiagonalization (JBD) process of a regular matrix pair $\{A,L\}$ is mathematically equivalent to two simultaneous Lanczos bidiagonalization processes of the upper and lower parts of the Q-factor of QR factorization of the stacked matrix $(A^{\mathrm T},\,L^{\mathrm T})^{\mathrm T}$ when their starting vectors are closely related in a specific way. The resulting JBD method for computing extreme generalized singular values and corresponding left and right singular vectors of $\{A,L\}$ realizes the standard Rayleigh--Ritz projection of the generalized singular value decomposition (GSVD) problem of $\{A,L\}$ onto the two left and one right subspaces generated by the JBD process. In this paper, the implicit restarting technique is nontrivially and skillfully extended to the JBD process, and an implicitly restarted JBD (IRJBD) algorithm is developed with proper selection of crucial shifts proposed and a few key implementation details addressed in finite precision arithmetic. Compact upper bounds are established for the residual norm of an approximate GSVD component in both exact and finite precision arithmetic, which are used to design efficient and reliable stopping criteria and avoid the expensive computation of approximate right generalized singular vectors. Numerical experiments illustrate that IRJBD performs well and is more efficient than the thick-restart JBD algorithm.