Automatic rational approximation and linearization of nonlinear eigenvalue problems

Pieter Lietaert, Karl Meerbergen, Javier Pérez, Bart Vandereycken

Research output: Contribution to journalArticlepeer-review

Abstract

We present a method for solving nonlinear eigenvalue problems (NEPs) using rational approximation. The method uses the Antoulas-Anderson algorithm (AAA) of Nakatsukasa, Sète and Trefethen to approximate the NEP via a rational eigenvalue problem. A set-valued variant of the AAA algorithm is also presented for building low-degree rational approximations of NEPs with a large number of nonlinear functions. The rational approximation is embedded in the state-space representation of a rational polynomial by Su and Bai. This procedure perfectly fits the framework of the compact rational Krylov methods (CORK and TS-CORK), allowing solve large-scale NEPs to be efficiently solved. One advantage of our method, compared to related techniques such as NLEIGS and infinite Arnoldi, is that it automatically selects the poles and zeros of the rational approximations. 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.

Original languageEnglish
Pages (from-to)1087-1115
Number of pages29
JournalIMA Journal of Numerical Analysis
Volume42
Issue number2
DOIs
StatePublished - Apr 1 2022

Keywords

  • nonlinear eigenvalue problem
  • rational Krylov method
  • rational interpolation

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