An Interpolating Polynomial Method for Numerical Conformal Mapping

A simple algorithm is presented for numerical approximation of conformal mappings for simply connected planar regions. The desired mapping is represented by a normalized polynomial of degree N, determined by the boundary values which it is presumed to take at the Nth roots of unity; then its values at the N intermediate 2Nth roots of unity are used to correct the initial guess. Each iteration, which consists essentially of matrix multiplication and boundary projection, costs O(N log N) arithmetic operations. Numerical results are provided to confirm linear convergence of the algorithm.