A fast implementation of a method to compute the Riemann mapping function is presented. The method has been recently introduced by N. Kerzman and the author; it expresses the Szegö kernel as the solution of an integral equation of the second kind. The complexity of this new algorithm is O(n² ), where n is the number of collocation points on the boundary of the region. Previous algorithms for mapping from the problem domain to the disk require O(n² log n) operations. It is shown how to treat symmetric regions. The algorithm is tested on several examples. The numerical results show that the method is competitive with respect to accuracy, stability, and efficiency.
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Manfred R. Trummer (1986) studied this question.
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