Characterizing GSVD by singular value expansion of linear operators and its computation

The generalized singular value decomposition (GSVD) of a matrix pair \A, L\ with A^m n and L^p n generalizes the singular value decomposition (SVD) of a single matrix. In this paper, we provide a new understanding of GSVD from the viewpoint of SVD, based on which we propose a new iterative method for computing nontrivial GSVD components of a large-scale matrix pair. By introducing two linear operators A and L induced by \A, L\ between two finite-dimensional Hilbert spaces and applying the theory of singular value expansion (SVE) for linear compact operators, we show that the GSVD of \A, L\ is nothing but the SVEs of A and L. This result characterizes completely the structure of GSVD for any matrix pair with the same number of columns. As a direct application of this result, we generalize the standard Golub-Kahan bidiagonalization (GKB) that is a basic routine for large-scale SVD computation such that the resulting generalized GKB (gGKB) process can be used to approximate nontrivial extreme GSVD components of \A, L\, which is named the gGKB\GSVD algorithm. We use the GSVD of \A, L\ to study several basic properties of gGKB and also provide preliminary results about convergence and accuracy of gGKB\GSVD for GSVD computation. Numerical experiments are presented to demonstrate the effectiveness of this method.