Fast computation of the circular map

This paper presents a new numerical implementation of Koebe's iterative method for computing the circular map of bounded and unbounded multiply connected regions of connectivity $m$. The computational cost of the method is $O(mn\ln n)$ where $n$ is the number of nodes in the discretization of each boundary component. The accuracy and efficiency of the presented method are demonstrated by several numerical examples. These examples include regions with high connectivity, regions whose boundaries are closer together, and regions with piecewise smooth boundaries.