Avoiding Discretization Issues for Nonlinear Eigenvalue Problems
Matthew J. Colbrook, Alex Townsend · SIAM Journal on Matrix Analysis and Applications · 2025
Abstract. The first step when solving an infinite-dimensional eigenvalue problem is often to discretize it. We show that one must be extremely careful when discretizing nonlinear eigenvalue problems. Using examples from the NLEVP collection, we demonstrate that discretization can lead to several issues, including (1) introduction of spurious eigenvalues, (2) omission of spectra, (3) severe ill-conditioning, and (4) emergence of ghost essential spectra. While many eigensolvers are available for solving finite matrix nonlinear eigenvalue problems, we propose InfBeyn, a solver for general holomorphic infinite-dimensional nonlinear eigenvalue problems that circumvents these discretization issues. We prove that InfBeyn is stable and converges. Furthermore, we provide an algorithm that computes the problem’s pseudospectra with explicit error control, enabling verification of computed spectra. Both algorithms and numerical examples are publicly available in the infNEP software package, which is written in MATLAB. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/MColbrook/infNEP and in the supplementary materials ( infNEP-main.zip [24.9KB]). [Formula: see text]