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：我们提出一种随机化、无逆的算法，用于生成任意 $n \times n$ 矩阵束 $(A,B)$ 的近似对角化。该算法的核心基于Ballard、Demmel与Dumitriu [Technical Report 2010] 最初提出的广义特征值问题的随机化分治特征求解器。我们证明，只要输入矩阵束具有足够良好的性质，这种分治方法能以高概率成功实现，这一结论通过推广Banks、Garza-Vargas、Kulkarni与Srivastava [Foundations of Computational Mathematics 2022] 最近的伪谱粉碎工作而达成。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，使得分治算法能在简单随机网格上运行，从而在向后误差意义上生成 $(A,B)$ 的精确对角化。本文的主要结果证明存在一种随机化算法，能以高概率（在精确算术下）生成可逆矩阵 $S,T$ 与对角矩阵 $D$，使得 $||A - SDT^{-1}||_2 \leq \varepsilon$ 且 $||B - ST^{-1}||_2 \leq \varepsilon$，且至多需要 $O \left(\log^2 \left( \frac{n}{\varepsilon} \right) T_{\text{MM}}(n) \right)$ 次运算，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新的保证，而且将近矩阵乘法时间确立为无逆、精确算术矩阵束对角化复杂度的上界。