Resolution of singularities by rational functions

TL;DR

This paper enhances the lightning method for rational approximation of functions with singularities by augmenting it with polynomial bases or poles at infinity, significantly improving accuracy and convergence rates.

Contribution

It introduces a robust augmentation of the lightning method, achieving optimal convergence rates for approximating functions like x^α on [0,1].

Findings

Enhanced approximation accuracy and convergence rate.

Achieved Stahl's optimal convergence rate via least-squares fitting.

Robust method combining lightning approximation with polynomial or infinite poles.

Abstract

Results on the rational approximation of functions containing singularities are presented. We build further on the ''lightning method'', recently proposed by Trefethen and collaborators, based on exponentially clustering poles close to the singularities. Our results are obtained by augmenting the lightning approximation set with either a low-degree polynomial basis or poles clustering towards infinity, in order to obtain a robust approximation of the smooth behaviour of the function. This leads to a significant increase in the achievable accuracy as well as the convergence rate of the numerical scheme. For the approximation of $x^\alpha$ on $[0,1]$, the optimal convergence rate as shown by Stahl in 1993 is now achieved simply by least-squares fitting.