Interpolation constrained rational minimax approximation with barycentric representation
For researchers in numerical approximation, this method offers a more stable and accurate approach to interpolation-constrained rational minimax approximation, though it is an incremental improvement over existing Lawson-type methods.
This paper introduces b-d-Lawson, a dual-based Lawson's method for rational minimax approximation under interpolation constraints, using barycentric representation to improve numerical stability and accuracy. Numerical results demonstrate its effectiveness.
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.