基于增广矩阵的CJ-FEAST SVD求解器，用于计算给定区间内奇异值的部分奇异值分解
An augmented matrix-based CJ-FEAST SVDsolver for computing a partial singular value decomposition with the singular values in a given interval
AI总结：
本文提出一种基于增广矩阵的CJ-FEAST SVD求解器，通过子空间迭代和谱投影计算给定区间内的奇异三元组，证明其数值后向稳定，并给出与叉积方法的收敛比较及选择策略。
摘要：
作者先前提出的基于叉积矩阵的CJ-FEAST SVD求解器在计算左奇异向量时可能比右奇异向量精度低得多，并且当期望的奇异值较小时可能在数值上后向不稳定。本文考虑一种替代的基于增广矩阵的CJ-FEAST SVD求解器，用于计算大型矩阵$A$的奇异三元组，其中奇异值位于包含在奇异谱中的区间$[a,b]$内。新的CJ-FEAST SVD求解器是将子空间迭代应用于增广矩阵$[0, A^T; A, 0]$的近似谱投影算子，该投影算子与$[a,b]$内的特征值相关联，并独立地构造具有期望奇异值的近似左、右奇异子空间，将$A$投影到这些子空间上以获得期望奇异三元组的Ritz近似。给出了近似谱投影算子精度的紧凑估计，并建立了若干收敛性结果。新求解器被证明在数值上始终后向稳定。对叉积和基于增广矩阵的CJ-FEAST SVD求解器进行了收敛性比较，并提出了一种通用的选择策略，以兼顾鲁棒性和整体效率。数值实验证实了所有结果。
英文摘要：
The cross-product matrix-based CJ-FEAST SVDsolver proposed previously by the authors is shown to compute the left singular vector possibly much less accurately than the right singular vector and may be numerically backward unstable when a desired singular value is small. In this paper, an alternative augmented matrix-based CJ-FEAST SVDsolver is considered to compute the singular triplets of a large matrix $A$ with the singular values in an interval $[a,b]$ contained in the singular spectrum. The new CJ-FEAST SVDsolver is a subspace iteration applied to an approximate spectral projector of the augmented matrix $[0, A^T; A, 0]$ associated with the eigenvalues in $[a,b]$, and constructs approximate left and right singular subspaces with the desired singular values independently, onto which $A$ is projected to obtain the Ritz approximations to the desired singular triplets. Compact estimates are given for the accuracy of the approximate spectral projector, and a number of convergence results are established. The new solver is proved to be always numerically backward stable. A convergence comparison of the cross-product and augmented matrix-based CJ-FEAST SVDsolvers is made, and a general-purpose choice strategy between the two solvers is proposed for the robustness and overall efficiency. Numerical experiments confirm all the results.
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