This paper explores a key question in numerical linear algebra: how can we compute projectors onto the deflating subspaces of a regular matrix pencil $(A,B)$, in particular without using matrix inversion or defaulting to an expensive Schur decomposition? We focus specifically on spectral projectors, whose associated deflating subspaces correspond to sets of eigenvalues/eigenvectors. In this work, we present a high-level approach to computing these projectors, which combines rational function approximation with an inverse-free arithmetic of Benner and Byers [Numerische Mathematik 2006]. The result is a numerical framework that captures existing inverse-free methods, generates an array of new options, and provides straightforward tools for pursuing efficiency on structured problems (e.g., definite pencils). To exhibit the efficacy of this framework, we consider a handful of methods in detail, including Implicit Repeated Squaring and iterations based on the matrix sign function. For the former, we present a new and general floating-point stability bound that may be of independent interest. In an appendix, we demonstrate that recent, randomized divide-and-conquer eigensolvers -- which are built on fast methods for individual projectors like those considered here -- can be adapted to produce the generalized Schur form of any matrix pencil in nearly matrix multiplication time.

翻译：本文探讨数值线性代数中的一个关键问题：如何计算正则矩阵束$(A,B)$的收缩子空间上的投影算子，特别是避免使用矩阵求逆或依赖昂贵的舒尔分解？我们特别关注谱投影算子，其关联的收缩子空间对应于特征值/特征向量集合。本工作提出一种计算这些投影算子的高层次方法，该方法将有理函数逼近与Benner和Byers提出的无逆算术方法[Numerische Mathematik 2006]相结合。由此产生的数值框架不仅涵盖了现有的无逆方法，还生成了一系列新方案，并为结构化问题（如定矩阵束）提供了直接的效率优化工具。为展示该框架的有效性，我们详细研究了若干方法，包括隐式重复平方和基于矩阵符号函数的迭代算法。针对前者，我们提出了一个新颖且通用的浮点稳定性界，该结果可能具有独立的研究价值。在附录中，我们论证了近期基于随机化分治的特征求解器——这类求解器建立在如本文所讨论的快速个体投影算子方法之上——可被推广至在接近矩阵乘法时间复杂度内计算任意矩阵束的广义舒尔形式。