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、无需矩阵求逆的算法，用于实现任意$n \times n$矩阵束$(A,B)$的近似对角化。该算法的主体基于Ballard、Demmel和Dumitriu [Technical Report 2010] 最初提出的广义特征值问题随机分治特征求解器。我们证明，只要输入矩阵束足够良态，这种分治方法就能以高概率成功实现，这通过推广Banks、Garza-Vargas、Kulkarni和Srivastava [Foundations of Computational Mathematics 2022] 近期关于伪谱破碎的工作来完成。特别地，我们表明对$(A,B)$进行扰动和缩放可正则化其伪谱，从而允许分治方法在一个简单随机网格上运行，并在向后误差意义下产生$(A,B)$的精确对角化。本文主要结果指出存在一种随机算法，能以高概率（且在精确算术下）得到可逆矩阵$S,T$和对角矩阵$D$，使得在至多$O \left( \log(n) \log^2 \left( \frac{n}{\varepsilon} \right) T_{\text{MM}}(n) \right)$次运算内满足$||A - SDT^{-1}||_2 \leq \varepsilon$和$||B - SIT^{-1}||_2 \leq \varepsilon$，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的保证，还确立了近乎矩阵乘法时间作为精确算术矩阵束对角化复杂度的上界。