A randomized FEAST algorithm for generalized eigenvalue problems

TL;DR

This paper introduces a new non-Hermitian FEAST algorithm for generalized eigenvalue problems, providing a stable, accurate, and convergent method that extends the original Hermitian-focused approach to broader cases.

Contribution

We develop a novel non-Hermitian FEAST algorithm with a new mathematical framework and convergence analysis, expanding its applicability beyond Hermitian problems.

Findings

The new algorithm successfully computes eigenvalues for non-Hermitian problems.

Numerical experiments confirm the effectiveness and convergence of the proposed method.

The method demonstrates stability and accuracy in complex eigenvalue regions.

Abstract

The FEAST algorithm, due to Polizzi, is a typical contour-integral based eigensolver for computing the eigenvalues, along with their eigenvectors, inside a given region in the complex plane. It was formulated under the circumstance that the considered eigenproblem is Hermitian. The FEAST algorithm is stable and accurate, and has attracted much attention in recent years. However, it was observed that the FEAST algorithm may fail to find the target eigenpairs when applying it to the non-Hermitian problems. Efforts have been made to adapt the FEAST algorithm to non-Hermitian cases. In this work, we develop a new non-Hermitian scheme for the FEAST algorithm. The mathematical framework will be established, and the convergence analysis of our new method will be studied. Numerical experiments are reported to demonstrate the effectiveness of our method and to validate the convergence properties.