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 as long as 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 $||B - SIT^{-1}||_2 \leq \varepsilon$ in at most $O \left( \log(n) \log^2 \left( \frac{n}{\varepsilon} \right) T_{\text{MM}}(n) \right)$ 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 exact arithmetic matrix pencil diagonalization.

翻译：我们提出一种随机化、无逆算法，用于实现任意$n \times n$矩阵束$(A,B)$的近似对角化。该算法主体基于Ballard、Demmel和Dumitriu [Technical Report 2010] 提出的广义特征值问题随机化分治求解器。我们证明，只要输入矩阵束充分良态，该分治方法即可通过推广Banks、Garza-Vargas、Kulkarni和Srivastava [Foundations of Computational Mathematics 2022] 近期提出的伪谱破碎工作，以高概率成功实现。具体而言，我们表明对$(A,B)$进行扰动与缩放可正则化其伪谱，从而允许分治算法在简单随机网格上运行，进而以向后误差意义生成$(A,B)$的精确对角化。本文主要结果证明：存在一种随机化算法，在精确算术条件下以高概率产生可逆矩阵$S,T$及对角矩阵$D$，使得$||A - SDT^{-1}||_2 \leq \varepsilon$且$||B - SIT^{-1}||_2 \leq \varepsilon$，其操作复杂度至多为$O \left( \log(n) \log^2 \left( \frac{n}{\varepsilon} \right) T_{\text{MM}}(n) \right)$，其中$T_{\text{MM}}(n)$为矩阵乘法的渐近复杂度。该结果不仅为高度并行的广义特征值求解器提供了新保证，还确立了精确算术矩阵束对角化复杂度的上界接近矩阵乘法时间。