A convex dual problem for the rational minimax approximation and Lawson's iteration

TL;DR

This paper introduces a convex programming approach for computing the rational minimax approximation in the complex plane, guaranteeing optimality under Ruttan's condition and improving convergence with a new Lawson's iteration.

Contribution

It proposes a convex optimization framework for rational minimax approximation and a novel Lawson's iteration that ensures convergence and verifiability.

Findings

Convex programming guarantees the rational minimax approximation under Ruttan's condition.

The new Lawson's iteration converges monotonically in numerical experiments.

The approach is competitive with existing methods like AAA and RKFIT.

Abstract

Computing the discrete rational minimax approximation in the complex plane is challenging. Apart from Ruttan's sufficient condition, there are few other sufficient conditions for global optimality. The state-of-the-art rational approximation algorithms, such as the adaptive Antoulas-Anderson (AAA), AAA-Lawson, and the rational Krylov fitting (RKFIT) method, perform highly efficiently, but the computed rational approximations may not be minimax solutions. In this paper, we propose a convex programming approach, the solution of which is guaranteed to be the rational minimax approximation under Ruttan's sufficient condition. Furthermore, we present a new version of Lawson's iteration for solving this convex programming problem. The computed solution can be easily verified as the rational minimax approximation. Our numerical experiments demonstrate that this updated version of Lawson's iteration generally converges monotonically with respect to the objective function of the convex optimization. It is an effective competitive approach for computing the rational minimax approximation, compared to the highly efficient AAA, AAA-Lawson, and the stabilized Sanathanan-Koerner iteration.