We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any \(n \times n\) matrix pencil (A, B). The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel and Dumitriu (Technical Report 2010). We demonstrate that this divide-and-conquer approach can be formulated to succeed with high probability provided the input pencil is sufficiently well-behaved, which is accomplished by generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni and Srivastava (Foundations of Computational Mathematics 2023). In particular, we show that perturbing and scaling (A, B) regularizes its pseudospectra, allowing divide-and-conquer to run over a simple random grid and in turn producing an accurate diagonalization of (A, B) in the backward error sense. The main result of the paper states the existence of a randomized algorithm that with high probability (and in exact arithmetic) produces invertible S, T and diagonal D such that \(||A - SDT^{-1}||_2 \le \varepsilon \) and \(||B - ST^{-1}||_2 \le \varepsilon \) in at most \(O \left( \log ^2 \left( \frac{n}{\varepsilon } \right) T_{\text {MM}}(n) \right) \) operations, where \(T_{\text {MM}}(n)\) is the asymptotic complexity of matrix multiplication. This not only provides a new set of guarantees for highly parallel generalized eigenvalue solvers but also establishes nearly matrix multiplication time as an upper bound on the complexity of inverse-free, exact-arithmetic matrix pencil diagonalization.

我们提出了一种随机的、无逆的算法，用于生成任何 \（n \times n\） 矩阵铅笔 （A， B） 的近似对角化。该算法的大部分内容都依赖于 Ballard、Demmel 和 Dumitriu 最初提出的广义特征值问题的随机分而治之的特征求解器（技术报告 2010）。我们证明，只要输入铅笔表现得足够好，这种分而治之的方法就可以以高概率成功，这是通过推广 Banks、Garza-Vargas、Kulkarni 和 Srivastava 最近的伪光谱粉碎工作来实现的（计算数学基础 2023）。特别是，我们表明扰动和缩放 （A， B） 使其伪光谱正则化，允许分而治之在简单的随机网格上运行，进而在向后误差意义上产生 （A， B） 的准确对角化。该论文的主要结果指出了一种随机算法的存在，该算法以高概率（并且在精确算术中）产生可逆的 S、T 和对角线 D，使得 \（||A - SDT^{-1}||_2 \le \varepsilon \） 和 \（||B - ST^{-1}||_2 \le \varepsilon \） 最多在 \（O \left（ \log ^2 \left（ \frac{n}{\varepsilon } \right） T_{\text {MM}}（n） \right） \） 运算中，其中 \（T_{\text {MM}}（n）\） 是矩阵乘法的渐近复杂度。这不仅为高度并行的广义特征值求解器提供了一组新的保证，而且还将近矩阵乘法时间确立为无逆精确算术矩阵铅笔对角化复杂度的上限。