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：针对复平面上解析函数（其奇点靠近插值区域，且区域为紧集，可能不连通或多连通），提出了一种具有特定指数收敛速率的新型重心插值算法。该方法的核心是通过求解Symm积分方程，利用逼近平衡对数势的离散分布高效计算插值节点与极点，并分别针对孤立奇点和分支点采用不同的极点分布策略。特别地，若不考虑极点，该方法可导出具有指数收敛性的多项式插值。数值实验验证了所提方法的优越性能。