NEP_MiniMax: An Approach for NEPs Based on Matrix-valued Minimax Approximations

TL;DR

NEP_MiniMax introduces a new method for solving nonlinear eigenvalue problems by combining rational minimax approximation with structure-exploiting linearization, achieving high accuracy and efficiency.

Contribution

The paper presents NEP_MiniMax, a novel approach that constructs matrix-valued rational approximations for NEPs and solves resulting polynomial eigenvalue problems, enhancing accuracy and computational performance.

Findings

Demonstrates competitiveness with state-of-the-art methods in benchmarks

Provides theoretical error bounds relating approximation quality to eigenpair accuracy

Achieves uniform accuracy with pole-free rational approximations in the domain

Abstract

We propose NEP_MiniMax, a novel computational method for solving nonlinear eigenvalue problems (NEPs) $T(\lambda)\mathbf{u}= 0$ on compact continua $\Omega \subset \mathbb{C}$. The method combines two key components: (1) a rational minimax approximation scheme where the {m-d-Lawson} algorithm constructs a minimax rational approximation for the vector-valued function from $T(x)$'s split form, yielding a matrix-valued rational approximation $R^*(x) = P^*(x)/q^*(x) \approx T(x)$, and (2) a structure-exploiting linearization technique. The minimax approximation guarantees uniform accuracy while generally keeping $R^*(x)$ pole-free in $\Omega$. Eigenpairs are then computed by solving a polynomial eigenvalue problem $P^*(\lambda) \mathbf{u}= 0$ via a strong linearization that exactly preserves eigenvalue multiplicities. Numerical experiments on benchmarks from the NLEVP collection demonstrate competitiveness with state-of-the-art methods (e.g., Beyn, NLEIGS, SV-AAA) in efficiency and accuracy, with theoretical error bounds directly relating eigenpair approximations to the rational approximation quality.