A Generalized Randomized Rank-Revealing Factorization

Grey Ballard, James Demmel, Ioana Dumitriu, Alexander Rusciano

Abstract: We introduce a Generalized Randomized QR-decomposition that may be applied to arbitrary products of matrices and their inverses, without needing to explicitly compute the products or inverses. This factorization is a critical part of a communication-optimal spectral divide-and-conquer algorithm for the nonsymmetric eigenvalue problem. In this paper, we establish that this randomized QR-factorization satisfies the strong rank-revealing properties. We also formally prove its stability, making it suitable in applications. Finally, we present numerical experiments which demonstrate that our theoretical bounds capture the empirical behavior of the factorization.