Randomized Implicitly Restarted Arnoldi method for the non-symmetric eigenvalue problem

TL;DR

This paper presents a novel randomized algorithm called rIRA for efficiently solving non-symmetric eigenvalue problems, combining sketching techniques with the Arnoldi process to improve computational performance while maintaining accuracy.

Contribution

It introduces a new randomized Implicitly Restarted Arnoldi method that retains key properties of IRA and enhances efficiency through sketching and restarting schemes.

Findings

The method effectively computes eigenvalues with reduced computational cost.

Numerical experiments demonstrate improved efficiency over traditional IRA.

The approach maintains accuracy while accelerating the eigenvalue computation.

Abstract

In this paper, we introduce a randomized algorithm for solving the non-symmetric eigenvalue problem, referred to as randomized Implicitly Restarted Arnoldi (rIRA). This method relies on using a sketch-orthogonal basis during the Arnoldi process while maintaining the Arnoldi relation and exploiting a restarting scheme to focus on a specific part of the spectrum. We analyze this method and show that it retains useful properties of the Implicitly Restarted Arnoldi (IRA) method, such as restarting without adding errors to the Ritz pairs and implicitly applying polynomial filtering. Experiments are presented to validate the numerical efficiency of the proposed randomized eigenvalue solver.