TY - JOUR
T1 - Automatic rational approximation and linearization of nonlinear eigenvalue problems
AU - Lietaert, Pieter
AU - Meerbergen, Karl
AU - Pérez, Javier
AU - Vandereycken, Bart
N1 - Publisher Copyright:
© 2021 The Author(s) 2021. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
PY - 2022/4/1
Y1 - 2022/4/1
N2 - 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.
AB - 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.
KW - nonlinear eigenvalue problem
KW - rational Krylov method
KW - rational interpolation
UR - http://www.scopus.com/inward/record.url?scp=85129392314&partnerID=8YFLogxK
U2 - 10.1093/imanum/draa098
DO - 10.1093/imanum/draa098
M3 - Article
AN - SCOPUS:85129392314
SN - 0272-4979
VL - 42
SP - 1087
EP - 1115
JO - IMA Journal of Numerical Analysis
JF - IMA Journal of Numerical Analysis
IS - 2
ER -