A novel barycentric interpolation algorithm with a specific exponential convergence rate is designed for analytic functions defined on the complex plane, with singularities located near the interpolation region, where the region is compact and can be disconnected or multiconnected. The core of the method is the efficient computation of the interpolation nodes and poles using discrete distributions that approximate the equilibrium logarithmic potential, achieved by solving a Symm's integral equation. It takes different strategies to distribute the poles for isolated singularities and branch points, respectively. In particular, if poles are not considered, it derives a polynomial interpolation with exponential convergence. Numerical experiments illustrate the superior performance of the proposed method.

翻译：本文设计了一种新的重心插值算法，适用于定义在复平面上、奇异点接近插值区域的解析函数插值问题。插值区域可以是紧凑的、可以是不连通或多连通的。该方法的核心是利用近似平衡对数势的离散分布来有效计算插值结点和极点，通过求解 Symm 积分方程实现。对于独立的奇异点和分支点，它采用不同的分布策略。特别地，如果不考虑极点，可推出具有指数收敛的多项式插值。数值实验说明了该方法的优越性。