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

Abstract

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.

Figures (8)

• Figure 1.1: Framework of NEP_MiniMax for solving NEPs based on m-d-Lawson and linearization.
• Figure 6.1: The left panel displays eigenvalue computations for \ref{['eq:example1']}, where blue and red dots represent eigenvalues inside and outside the sampled region (black dots) respectively, obtained via Algorithm \ref{['alg:linear_mdlawson']}; original eigenvalues from \ref{['eq:example1']} are shown as blue circles for comparison. The right panel plots the poles of ${\boldsymbol{\xi}}^*$ as triangles, demonstrating that m-d-Lawson produces a numerically best rational approximation---notably pole-free within our region of interest.
• Figure 6.2: Left panel: Computed eigenvalues of \ref{['eq:time_delay2_def']} obtained via NEP_MiniMax ($+$), Beyn's method ($\circ$), and SV-AAA ($\times$). Right panel: Illustration of poles of $\boldsymbol{\xi}^*$ ($\bigtriangleup$) and zeros of $p^*_1$ ($+$), $p^*_2$ ($\square$), and $p^*_3$ ($\Diamond$), computed using m-d-Lawson. Only points with magnitude less than 18 are displayed.
• Figure 6.3: Left: Eigenvalues of \ref{['eq:hadeler_def']} computed via NEP_MiniMax ($\circ$), Beyn's method ($+$), SV-AAA ($\times$), and RSI ($\square$) are displayed. Right: Relative residuals $\epsilon(\lambda,\boldsymbol{u})$ of the eigenpairs obtained from NEP_MiniMax, Beyn's method, SV-AAA, and RSI in Example \ref{['ex:hadeler']}.
• Figure 6.4: Left: Actual relative residuals of the eigenpairs computed by NEP_MiniMax using rational approximations of degrees 6, 7, and 8, along with their corresponding a priori estimates $\sqrt{\|\mathscr{G}_{\{E_i\}_i}\|_2 \, e(\boldsymbol{\xi}^*)}$. Here, NEP_MiniMax-6 refers to the computation where m-d-Lawson produces a type (6,6) rational approximation, followed by linearization and eigenvalue extraction (priori6 denotes the a priori estimate for NEP_MiniMax-6; analogous conventions apply to degrees 7 and 8). Right: Evolution of residuals $\epsilon(\lambda, \boldsymbol{u})$ for eigenpairs in $\Omega$ over the first 10 iterations of NEP_MiniMax-FILTER-6.

Theorems & Definitions (28)

• Remark 2.1
• Proposition 2.1: zhzz:2025
• Proposition 2.2
• Proposition 2.3
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Corollary 3.1
• proof