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 $\{(x_{\mathrm{\ell }},F(x_{\mathrm{\ell }})){\}}_{\mathrm{\ell }=1}^m$ 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 $R(x)=P(x)\mathrm{/}q(x)$ (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: $$\underset{R(x)=P(x)\mathrm{/}q(x)}{\inf }\underset{1\le \mathrm{\ell }\le m}{\max }\mathrm{\parallel }F(x_{\mathrm{\ell }})-R(x_{\mathrm{\ell }}){\mathrm{\parallel }}_{\mathrm{F}}\mathrm{.}$$ 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 ($\mathsf{\text{m-d-Lawson}}$) to solve the dual problem, generalizing Lawson’s iteration to matrix-valued functions. Convergence analysis of $\mathsf{\text{m-d-Lawson}}$ 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.