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 2022]. 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 $||A - SDT^{-1}||_2 \leq \varepsilon$ and $||B - ST^{-1}||_2 \leq \varepsilon$ 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.

翻译：本文提出一种随机化、无逆算法，用于近似对角化任意 $n \times n$ 矩阵束 $(A,B)$。该算法的主体基于Ballard、Demmel和Dumitriu [技术报告2010] 最初提出的广义特征值问题的随机化分治特征求解器。我们论证了若输入矩阵束充分良态，则该分治方法可被构造为高概率成功，这一结论通过推广Banks、Garza-Vargas、Kulkarni和Srivastava [计算数学基础 2022] 近期提出的伪谱碎裂工作得以实现。特别地，我们表明对 $(A,B)$ 进行扰动和缩放可正则化其伪谱，使分治法能在简单随机网格上运行，从而在后向误差意义下生成 $(A,B)$ 的精确对角化。本文主要结果证明了随机化算法的存在性：在高概率下（使用精确算术），可生成可逆矩阵 $S,T$ 及对角矩阵 $D$，使得 $||A - SDT^{-1}||_2 \leq \varepsilon$ 且 $||B - ST^{-1}||_2 \leq \varepsilon$，其运算复杂度不超过 $O \left(\log^2 \left( \frac{n}{\varepsilon} \right) T_{\text{MM}}(n) \right)$，其中 $T_{\text{MM}}(n)$ 为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新保证，更确立了近矩阵乘法时间作为无逆精确算术矩阵束对角化复杂度的上界。