Automatic rational approximation and linearization of nonlinear eigenvalue problems

TL;DR

This paper introduces an efficient, automatic rational approximation method for solving large-scale nonlinear eigenvalue problems, outperforming existing techniques in accuracy and computational effort.

Contribution

It develops a novel, automatic rational approximation approach using AAA and state space embedding, improving efficiency and simplicity over prior methods like NLEIGS.

Findings

Method is competitive with NLEIGS in accuracy.

Produces smaller linearizations with less user effort.

Efficiently solves large-scale nonlinear eigenvalue problems.

Abstract

We present a method for solving nonlinear eigenvalue problems using rational approximation. The method uses the AAA method by Nakatsukasa, S\`{e}te, and Trefethen to approximate the nonlinear eigenvalue problem by a rational eigenvalue problem and is embedded in the state space representation of a rational polynomial by Su and Bai. The advantage of the method, compared to related techniques such as NLEIGS and infinite Arnoldi, is the efficient computation by an automatic procedure. In addition, a set-valued approach is developed that allows building a low degree rational approximation of a nonlinear eigenvalue problem. The method perfectly fits the framework of the Compact rational Krylov methods (CORK and TS-CORK), allowing to efficiently solve large scale nonlinear eigenvalue problems. Numerical examples show that the presented framework is competitive with NLEIGS and usually produces smaller linearizations with the same accuracy but with less effort for the user.