NEP_MiniMax: An Approach for NEPs Based on Matrix-valued Minimax Approximations
This addresses computational challenges in solving NEPs for applications like physics and engineering, but it is incremental as it builds on existing approximation and linearization techniques.
The paper tackles nonlinear eigenvalue problems (NEPs) by proposing NEP_MiniMax, a method that combines rational minimax approximation and structure-exploiting linearization to compute eigenpairs with uniform accuracy and preserved multiplicities, demonstrating competitiveness with state-of-the-art methods in efficiency and accuracy on benchmarks.
We propose NEP_MiniMax, a novel computational method for solving nonlinear eigenvalue problems (NEPs) $T(λ)\mathbf{u}= 0$ on compact continua $Ω\subset \mathbb{C}$. 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 $Ω$. Eigenpairs are then computed by solving a polynomial eigenvalue problem $P^*(λ) \mathbf{u}= 0$ 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.