Generalized Pseudospectral Shattering and Inverse-Free Matrix Pencil Diagonalization

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 2023). 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 \(||A - SDT^{-1}||_2 \le \varepsilon \) and \(||B - ST^{-1}||_2 \le \varepsilon \) 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.