On Fornberg’s Numerical Method for Conformal Mapping

Abstract

It is shown that Fornberg’s method [1] is a discretized version of an iterative process which is essentially the same we introduced and studied in [7], [8]. The main difference consists in the determination of the shift $U_k $ in each iterative step. While we give an explicit representation of $U_k $, Fornberg determines $U_k $ from a linear equation, which involves a certain integral operator R. We prove some properties and estimates for this operator which explain several experimental results reported in [1], especially the fact that the conjugate gradient method is so efficient. The results of [8] concerning the convergence can be applied also for Fornberg’s method. Finally, we give estimates for the numerical error.