NEP_MiniMax: An approach for NEPs based on minimax approximation of the split coefficient vector

Abstract

We propose \textsf{NEP\_MiniMax}, a novel computational method for solving nonlinear eigenvalue problems (NEPs) $T(λ)\mathbf{u} = \mathbf{0}$ on a Jordan domain $Ω\subset \mathbb{C}$. For an NEP in split form $T(x)=\sum_{i=1}^s t_i(x)E_i$, the method applies the \textsf{m-d-Lawson} algorithm to the split coefficient vector-valued function $\mathbf{t}(x)=[t_1(x),\ldots,t_s(x)]^{\textrm{T}}$ on $Ω$, rather than directly minimizing an error for the matrix-valued function $T$. The resulting rational minimax approximant ${\boldsymbolξ}^*(x)=[r_1^*(x),\ldots,r_s^*(x)]^{\textrm{T}}$ induces the rational matrix surrogate $R^*(x)=\sum_{i=1}^s r_i^*(x)E_i=P^*(x)/q^*(x)\approx T(x)$. The method combines this coefficient-vector approximation with a structure-exploiting linearization technique. A matrix error bound transfers the uniform coefficient-vector approximation accuracy to $R^*$ and yields computable eigenpair residual estimates. Eigenpairs are then computed by solving a polynomial eigenvalue problem $P^*(λ) \mathbf{u} = \mathbf{0}$ via a strong linearization that exactly preserves eigenvalue multiplicities of $P^*$. Numerical experiments on some problems 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 {residuals} to the {split coefficient-vector approximation} quality.