arXiv subjects

Chenkun Zhang

Publications and source records attributed to Chenkun Zhang.

3 recordsLinked to original sources

arXiv2026

Rational Minimax Approximations for Matrix-Valued Functions: Existence, Optimality and Algorithms

In this paper, we study rational minimax approximation for continuous complex matrix-valued functions in the Frobenius norm, where all approximant entries share a common denominator. This generalizes classical scalar rational approximation, with applications in system modeling, microwave design, and nonlinear eigenvalue problems. We first prove the existence of such matrix-valued approximants on point sets dense in themselves, extending Walsh's foundational scalar result. Next, we establish characterizations of the local and global minimax approximants by deriving primal/dual matrix-valued Kolmogorov criteria and a Ruttan-type sufficient condition for global optimality. For analytic functions on a continuum, we link continuum minimax approximation to approximation on its boundary, and finite boundary samples via the maximum norm principle. We show that Ruttan's sufficient optimality condition provides a certificate under which a minimax approximant obtained from the boundary or from a discrete set of boundary nodes also solves the original continuum problem. Finally, for discrete approximation, we connect these conditions to a dual problem and the related dual-based numerical method m-d-Lawson: when the original minimax problem admits a solution, strong duality is equivalent to Ruttan's sufficient optimality condition, and, the optimality equations underlying the m-d-Lawson iteration coincide with Kolmogorov's dual criteria. These results provide a theoretical basis for certifying and computing matrix-valued rational minimax approximants.

math.OC↗

arXiv2026

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 $\Omega \subset \mathbb{C}$. 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 $R^*(x) = P^*(x)/q^*(x) \approx T(x)$, and (2) a structure-exploiting linearization technique. The minimax approximation guarantees uniform accuracy while generally keeping $R^*(x)$ pole-free in $\Omega$. Eigenpairs are then computed by solving a polynomial eigenvalue problem $P^*(\lambda) \mathbf{u}= 0$ 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.

math.NA↗

arXiv2025

Rational minimax approximation of matrix-valued functions

In this paper, we present a rigorous framework for rational minimax approximation of matrix-valued functions that generalizes classical scalar approximation theory. Given sampled data $\{(x_\ell, {F}(x_\ell))\}_{\ell=1}^m$ where ${F}:\mathbb{C} \to \mathbb{C}^{s \times t}$ is a matrix-valued function, we study the problem of finding a matrix-valued rational approximant ${R}(x) = {P}(x)/q(x)$ (with ${P}:\mathbb{C} \to \mathbb{C}^{s \times t}$ a matrix-valued polynomial and $q(x)$ a nonzero scalar polynomial of prescribed degrees) that minimizes the worst-case Frobenius norm error over the given nodes: $$ \inf_{{R}(x) = {P}(x)/q(x)} \max_{1 \leq \ell \leq m} \|{F}(x_\ell) - {R}(x_\ell)\|_{\rm F}. $$ By reformulating this min-max optimization problem through Lagrangian duality, we derive a maximization dual problem over the probability simplex. We analyze weak and strong duality properties and establish a sufficient condition ensuring that the solution of the dual problem yields the minimax approximant $R(x)$. For numerical implementation, we propose an efficient method (\textsf{m-d-Lawson}) to solve the dual problem, generalizing Lawson's iteration to matrix-valued functions. Convergence analysis of \textsf{m-d-Lawson} is established. Numerical experiments are conducted and compared to state-of-the-art approaches, demonstrating its efficiency as a novel computational framework for matrix-valued rational approximation.

math.NA↗