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.