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.

翻译：矩阵对$\{A, L\}$的广义奇异值分解（GSVD）推广了单个矩阵的奇异值分解（SVD），其中$A\in\mathbb{R}^{m\times n}$，$L\in\mathbb{R}^{p\times n}$。本文从SVD视角提出对GSVD的新理解，并据此提出一种计算大规模矩阵对非平凡GSVD分量的迭代方法。通过引入由$\{A, L\}$诱导的两个有限维希尔伯特空间之间的线性算子$\mathcal{A}$与$\mathcal{L}$，并应用紧线性算子的奇异值展开（SVE）理论，我们证明$\{A, L\}$的GSVD本质上即为$\mathcal{A}$与$\mathcal{L}$的SVE。该结果完整刻画了任意具有相同列数的矩阵对的GSVD结构。基于此结论，我们推广了大规模SVD计算的基础算法——标准Golub-Kahan双对角化（GKB），使推广后的广义GKB（gGKB）过程能够逼近$\{A, L\}$的非平凡极值GSVD分量，并命名为gGKB_GSVD算法。利用$\{A, L\}$的GSVD研究了gGKB的若干基本性质，同时给出了gGKB_GSVD在GSVD计算中的收敛性与精度初步分析。数值实验验证了该方法的有效性。