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

We propose NEP_MiniMax, a novel computational method for solving nonlinear eigenvalue problems (NEPs) $T(\lambda )\mathbf{u}=0$ on compact continua $\mathrm{\Omega }\subset \mathbb{C}$. 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 $R^{\ast }(x)=P^{\ast }(x)\mathrm{/}{q}^{\ast }(x)\approx T(x)$, and (2) a structure-exploiting linearization technique. The minimax approximation guarantees uniform accuracy while generally keeping $R^{\ast }(x)$ pole-free in $\mathrm{\Omega }$. Eigenpairs are then computed by solving a polynomial eigenvalue problem $P^{\ast }(\lambda )\mathbf{u}=0$ 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.