SIAM Journal on Scientific Computing, Ahead of Print.
This paper describes a set of rational filtering algorithms to compute a few eigenvalues (and associated eigenvectors) of non-Hermitian matrix pencils. Our interest lies in computing eigenvalues located inside a given disk, and the proposed algorithms approximate these eigenvalues and associated eigenvectors by harmonic Rayleigh--Ritz projections on subspaces built by computing range spaces of rational matrix functions through randomized range finders. These rational matrix functions are designed so that directions associated with nonsought eigenvalues are dampened to (approximately) zero. Variants based on matrix partitionings are introduced to further reduce the overall complexity of the proposed framework. Compared with existing eigenvalue solvers based on rational matrix functions, the proposed technique requires no estimation of the number of eigenvalues located inside the disk. Several theoretical and practical issues are discussed, and the competitiveness of the proposed framework is demonstrated via numerical experiments.

中文翻译：

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基于有理滤波和矩阵划分的快速随机非厄米特征求解器

SIAM 科学计算杂志，提前印刷。
本文描述了一组有理滤波算法来计算非厄米矩阵柱的一些特征值（和相关的特征向量）。我们的兴趣在于计算位于给定磁盘内的特征值，并且所提出的算法通过在通过随机测距仪计算有理矩阵函数的范围空间而构建的子空间上的谐波瑞利-里兹投影来近似这些特征值和相关的特征向量。这些有理矩阵函数被设计成与非搜索特征值相关的方向被抑制到（大约）零。引入了基于矩阵分区的变体，以进一步降低所提出框架的整体复杂性。与现有的基于有理矩阵函数的特征值求解器相比，所提出的技术不需要估计位于磁盘内部的特征值的数量。讨论了几个理论和实践问题，并通过数值实验证明了所提出框架的竞争力。