A rational approximation method for the nonlinear eigenvalue problem

TL;DR

This paper introduces a rational approximation method for nonlinear eigenvalue problems that linearizes the problem, enabling efficient computation of eigenvalues in specific regions, suitable for large-scale engineering applications.

Contribution

It proposes a novel rational approximation technique that simplifies nonlinear eigenvalue problems and efficiently computes eigenvalues in targeted regions, with theoretical validation and numerical testing.

Findings

Effective in large-scale problems

Able to compute eigenvalues in specific regions

Simple implementation and computational efficiency

Abstract

This paper presents a method for computing eigenvalues and eigenvectors for some types of nonlinear eigenvalue problems. The main idea is to approximate the functions involved in the eigenvalue problem by rational functions and then apply a form of linearization. Eigenpairs of the expanded form of this linearization are not extracted directly. Instead, its structure is exploited to develop a scheme that allows to extract all eigenvalues in a certain region of the complex plane by solving an eigenvalue problem of much smaller dimension. Because of its simple implementation and the ability to work efficiently in large dimensions, the presented method is appealing when solving challenging engineering problems. A few theoretical results are established to explain why the new approach works and numerical experiments are presented to validate the proposed algorithm.