Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization

TL;DR

This paper introduces a randomized, inverse-free algorithm for approximate diagonalization of matrix pencils, leveraging pseudospectral shattering and divide-and-conquer methods to achieve high probability success with near matrix multiplication time complexity.

Contribution

It generalizes pseudospectral shattering to regularize matrix pencils, enabling an inverse-free, divide-and-conquer eigensolver with high probability guarantees and near matrix multiplication time complexity.

Findings

Algorithm produces approximate diagonalization in near matrix multiplication time.

Perturbation and scaling regularize pseudospectra for divide-and-conquer success.

Provides high-probability guarantees for inverse-free generalized eigenvalue computation.

Abstract

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.