Barycentric rational interpolation of exponentially clustered poles

Abstract

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.