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.

翻译：本文提出了一种新颖的重心插值算法，适用于定义在复平面上的解析函数 $f\in\mathcal{A}(E)$。该算法涵盖多项式与有理插值，专门用于处理 $E$ 附近的奇异性。我们的方法适用于由分段光滑若尔当曲线界定的区域 $E$，且对区域连通性无任何限制。本方法的核心特点在于通过数值求解 Symm 积分方程高效计算离散点，从而构建多项式或有理重心插值式。此外，本方法可提供平衡势的相关参数，如 Robin 常数，该参数可用于估计收敛速率。数值实验展示了本方法所实现的收敛速率与理论收敛速率的对比结果。