BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//CS-TR-79-710
       ENTRY:: June 19, 1995
ORGANIZATION:: Stanford University, Department of Computer Science
       TITLE:: Numerical computation of the Schwarz-Christoffel
               transformation
        TYPE:: Technical Report
      AUTHOR:: Trefethen, Lloyd N.
        DATE:: March 1979
       PAGES:: 58
    ABSTRACT:: A program is described which computes Schwarz-Christoffel
               transformations that map the unit disk conformally onto the
               interior of a bounded or unbouded polygon in the complex
               plane. The inverse map is also computed. The computational
               problem is approached by setting up a nonlinear system of
               equations whose unknowns are essentially the "accessory
               parameters" $z_k$. This system is then solved with a packaged
               subroutine.

               New features of this work include the evaluation of integrals
               within the disk rather than along the boundary, making
               possible the treatment of unbounded polygons; the use of a
               compound form of Gauss-Jacobi quadrature to evaluate the
               Schwarz-Christoffel integral, making possible high accuracy
               at reasonable cost; and the elimination of constraints in the
               nonlinear system by a simple change of variables.

               Schwarz-Christoffel transformations may be applied to solve
               the Laplace and Poisson equations and related problems in
               two-dimensional domains with irregular or unbounded (but not
               curved or multiply connected) geometries. Computational
               examples are presented. The time required to solve the
               mapping problem is roughly proportional to $N^3$, where N is
               the number of vertices of the polygon. A typical set of
               computations to 8-place accuracy with $N \leq 10$ takes 1 to
               10 seconds on an IBM 370/168.
       NOTES:: [Adminitrivia V1/Prg/19950619]
         END:: STAN//CS-TR-79-710