Stabilizing the Rayleigh--Ritz procedure by randomization

TL;DR

This paper introduces a randomized Rayleigh--Ritz procedure that improves convergence in eigenpair extraction from subspaces, addressing longstanding issues with the standard method.

Contribution

It proposes a novel randomized approach that ensures convergence rates comparable to ideal projections, applicable to both linear and nonlinear eigenvalue problems.

Findings

The randomized method achieves convergence rates similar to ideal projections.

It extends naturally to nonlinear eigenvalue problems.

The approach addresses long-standing convergence issues in eigenpair extraction.

Abstract

Extracting approximate eigenpairs from a prescribed subspace is of fundamental importance in eigenvalue computation. While projecting the target eigenvector onto the subspace yields satisfactory accuracy, extracting an approximate eigenpair that attains a comparable convergence rate has remained a long-standing open problem. Although the standard Rayleigh--Ritz procedure is widely used for this purpose, it may suffer from deteriorated convergence of Ritz values and may even fail to produce convergent Ritz vectors. In this paper, we address this long-standing open problem by introducing a randomized Rayleigh--Ritz procedure whose output converges at a rate similar to the ideal projection. Our analysis requires only the simplicity of the target eigenvalue and extends naturally to nonlinear eigenvalue problems.