Fast randomized non-Hermitian eigensolver based on rational filtering and matrix partitioning

TL;DR

This paper introduces a fast, randomized rational filtering method for efficiently computing a few interior eigenvalues of non-Hermitian matrices, leveraging matrix partitioning to reduce complexity without needing eigenvalue count estimates.

Contribution

The paper presents a novel rational filtering algorithm with matrix partitioning for non-Hermitian eigenproblems, eliminating the need for eigenvalue count estimation.

Findings

Demonstrates competitiveness through numerical experiments

Reduces computational complexity with matrix partitioning

Does not require prior eigenvalue count estimation

Abstract

This paper describes a set of rational filtering algorithms to compute a few eigenvalues (and associated eigenvectors) of non-Hermitian matrix pencils. Our interest lies in computing eigenvalues located inside a given disk, and the proposed algorithms approximate these eigenvalues and associated eigenvectors by harmonic Rayleigh-Ritz projections on subspaces built by computing range spaces of rational matrix functions through randomized range finders. These rational matrix functions are designed so that directions associated with non-sought eigenvalues are dampened to (approximately) zero. Variants based on matrix partitionings are introduced to further reduce the overall complexity of the proposed framework. Compared with existing eigenvalue solvers based on rational matrix functions, the proposed technique requires no estimation of the number of eigenvalues located inside the disk. Several theoretical and practical issues are discussed, and the competitiveness of the proposed framework is demonstrated via numerical experiments.