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_, F(x₎)\₌₁ᵐ where F:C → Cs × 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:C → Cs × 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) max1 ≤ ≤ m \|F(x₎ - R(x₎\|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 (m-d-Lawson) to solve the dual problem, generalizing Lawson's iteration to matrix-valued functions. Convergence analysis of 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.