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) $T(\lambda )\mathbf{u}=0$ on a Jordan domain $\Omega \subset \mathbb{C}$. 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 $\mathbf{t}(x)=[t_1(x),\dots ,t_s(x){]}^{\text{T}}$ on $\Omega$, rather than directly minimizing an error for the matrix-valued function $T$. The resulting rational minimax approximant ${\mathbf{\xi }}^{\ast }(x)=[r_1^{\ast }(x),\dots ,r_s^{\ast }(x){]}^{\text{T}}$ induces the rational matrix surrogate $R^{\ast }(x)={\sum }_{i=1}^s{r}_i^{\ast }(x)E_i=P^{\ast }(x)/q^{\ast }(x)\approx T(x)$. 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 $R^{\ast }$ and yields computable eigenpair residual estimates. 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 of $P^{\ast }$. 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.

arXiv comment45 pages