Barycentric Interpolation Based on Equilibrium Potential

TL;DR

This paper introduces a new barycentric interpolation method for complex analytic functions that efficiently handles singularities and provides convergence estimates, applicable to regions with complex boundaries.

Contribution

The paper presents a novel barycentric interpolation algorithm for complex functions, incorporating equilibrium potential parameters and solving Symm's integral equation for improved accuracy.

Findings

Demonstrates convergence rates matching theoretical predictions

Handles singularities near the region boundary effectively

Applicable to regions with piecewise smooth Jordan curves

Abstract

We present a novel barycentric interpolation algorithm designed for analytic functions $f\in\mathcal{A}(E)$ defined on the complex plane. The algorithm, which encompasses both polynomial and rational interpolation, is tailored to handle singularities near $E$. Our method is applicable to regions $E$ bounded by piecewise smooth Jordan curves, and it imposes no connectivity restrictions on the region. The key feature of our approach lies in efficiently computing discrete points via the numerical solution of Symm's integral equation, enabling the construction of polynomial or rational barycentric interpolants. Furthermore, our method provides relevant parameters for the equilibrium potential, such as Robin's constant, which can be used to estimate convergence rates. Numerical experiments demonstrate the convergence rate achieved by our method in comparison to the theoretical convergence rate.