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 $A\in\mathbb{R}^{m\times n}$ and $L\in\mathbb{R}^{p\times n}$ 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 $\mathcal{A}$ and $\mathcal{L}$ 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 $\mathcal{A}$ and $\mathcal{L}$. 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.