We have developed a rational interpolation method for analytic functions with branch point singularities, which utilizes several exponentially clustered poles proposed by Trefethen and his collaborators (2021, Exponential node clustering at singularities for rational approximation, quadrature, and PDEs. Numer. Math., 147, 227–254). The key to the feasibility of this interpolation method is that the interpolation nodes approximately satisfy the distribution of the equilibrium potential. These nodes make the convergence rate of the rational interpolation consistent with the theoretical rates, and steadily approach machine accuracy. The technique can be used, not only for the interval $[0,1]$, but can also be extended to include corner regions and the case of multiple singularities.

中文翻译：

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指数聚集极点的重心有理插值


我们开发了一种用于具有分支点奇点的解析函数的有理插值方法，该方法利用了 Trefethen 及其合作者提出的几个指数聚类极点（2021，Exponential node clustering at奇异点用于有理逼近、求积和偏微分方程。Numer.Math.，147 ，227-254）。该插值方法可行性的关键在于插值节点近似满足平衡势的分布。这些节点使得有理插值的收敛速度与理论速度一致，并稳定地接近机器精度。该技术不仅可以用于区间$[0,1]$，还可以扩展到包括角区域和多个奇点的情况。