Interpolation constrained rational minimax approximation with barycentric representation

TL;DR

This paper introduces a dual-based Lawson's method called b-d-Lawson for rational minimax approximation with interpolation constraints, improving stability and efficiency using barycentric representation and dual optimization.

Contribution

The paper presents a novel dual framework combined with barycentric rational functions to enhance stability and efficiency in interpolation-constrained minimax approximation.

Findings

Enhanced numerical stability over Vandermonde basis methods

Efficient transformation of min-max problems via dual framework

Numerical results demonstrate improved approximation performance

Abstract

In this paper, we propose a novel dual-based Lawson's method, termed {b-d-Lawson}, designed for addressing the rational minimax approximation under specific interpolation conditions. The {b-d-Lawson} approach incorporates two pivotal components that have been recently gained prominence in the realm of the rational approximations: the barycentric representation of the rational function and the dual framework for tackling minimax approximation challenges. The employment of barycentric formulae enables a streamlined parameterization of the rational function, ensuring natural satisfaction of interpolation conditions while mitigating numerical instability typically associated with Vandermonde basis matrices when monomial bases are utilized. This enhances both the accuracy and computational stability of the method. To address the bi-level min-max structure, the dual framework effectively transforms the challenge into a max-min dual problem, thereby facilitating the efficient application of Lawson's iteration. The integration of this dual perspective is crucial for optimizing the approximation process. We will discuss several applications of interpolation-constrained rational minimax approximation and illustrate numerical results to evaluate the performance of the {b-d-Lawson} method.