Fast barycentric rational interpolations for complex functions with some singularities

Based on Cauchy’s integral formula and conformal maps, this paper presents a new method for constructing barycentric rational interpolation formulae for complex functions, which may contain singularities such as poles, branch cuts, or essential singularities. The resulting interpolations are pole-free, exponentially convergent, and numerically stable, requiring only \({\mathcal {O}}(N)\) operations. Inspired by the logarithm equilibrium potential, we introduce a Möbius transform to concentrate nodes to the vicinity of singularity to get a spectacular improvement on approximation quality. A thorough convergence analysis is provided, alongside numerous numerical examples that illustrate the theoretical results and demonstrate the accuracy and efficiency of the methodology. Meanwhile, the paper also discusses some applications of the method including the numerical solutions of boundary value problems and the zero locations of holomorphic functions.