Numerical Experiments in Conformal Mapping by the Method of Orthonormal Polynomials

Conformal maps of simply connected regions onto the interior and exterior of the unit circle can be computed from complex polynomials orthonormal over the boundary of the given region. As an example of this method, maps of ellipses, rectangles, limacons and ovals of Cassini were computed on the IBM 7070. The results indicate that, in general, the method gives good accuracy, but roundoff error can become very serious, indicating the desirability of computing with greater precision. For fourfold symmetric regions, the algorithm can be simplified to decrease significantly the amount of computation resquired to achieve the same accuracy as that achieved by the algorithm for the general case.