We propose a contour integral-based algorithm for computing a few singular values of a matrix or a few generalized singular values of a matrix pencil. Mathematically, the generalized singular values of a matrix pencil are the eigenvalues of an equivalent Hermitian-definite matrix pencil, known as the Jordan-Wielandt matrix pencil. However, direct application of the FEAST solver does not fully exploit the structure of this problem. We analyze several projection strategies on the Jordan-Wielandt matrix pencil, and propose an effective and robust scheme tailored to GSVD. Both theoretical analysis and numerical experiments demonstrate that our algorithm achieves rapid convergence and satisfactory accuracy.

翻译：我们提出一种基于围道积分的算法，用于计算矩阵的若干奇异值或矩阵束的若干广义奇异值。从数学角度来看，矩阵束的广义奇异值等价于一个埃尔米特正定矩阵束（即约当-维兰德矩阵束）的特征值。然而，直接应用FEAST求解器无法充分利用该问题的结构。我们分析了作用于约当-维兰德矩阵束的若干投影策略，并提出了一种针对GSVD的有效且稳健的方案。理论分析与数值实验均表明，我们的算法能够实现快速收敛并达到令人满意的精度。