. Thanks to powerful algorithms and computers, Schwarz-- Christoffel mapping is a practical reality. With the ability to compute have come new mathematical ideas. The state of the art is surveyed. 1991 Mathematics Subject Classification: 30C30, 31A05 Keywords and Phrases: conformal mapping, Schwarz--Christoffel formula 1. Introduction. In the past twenty years, because of new algorithms and new computers, Schwarz--Christoffel conformal mapping of polygons has matured to a technology that can be used at the touch of a button. Many authors have contributed to this progress, including Dappen, Davis, Dias, Elcrat, Floryan, Henrici, Hoekstra, Howell, Hu, Reppe, Zemach, and ourselves. The principal SC software tools are the Fortran package SCPACK [15] and its more capable Matlab successor, the Schwarz--Christoffel Toolbox [3]. It is now a routine matter to compute an SC map involving a dozen vertices to ten digits of accuracy in a few seconds on a workstation. With the power to compute...

. Eigenvalues with the eigenvector condition number, the field of values, and pseudospectra have all been suggested as the basis for convergence bounds for minimum residual Krylov subspace methods applied to non-normal coefficient matrices. This paper analyzes and compares these bounds, illustrating with six examples the success and failure of each one. Refined bounds based on eigenvalues and the field of values are suggested to handle low-dimensional non-normality. It is observed that pseudospectral bounds can capture multiple convergence stages. Unfortunately, computation of pseudospectra can be rather expensive. This motivates an adaptive technique for estimating GMRES convergence based on approximate pseudospectra taken from the Arnoldi process that is the basis for GMRES. Key words. Krylov subspace methods, GMRES convergence, non-normal matrices, pseudospectra, field of values AMS subject classifications. 15A06, 65F10, 15A18, 15A60, 31A15 1. Introduction. Popular algorithms for...

Motivated both by digital filter design and polynomial-based matrix iteration methods, we study Green's function for the complement of a union of disjoint closed intervals. The key tool is the Schwarz--Christoffel map. Asymptotic analysis produces simple and useful leading terms for Green's function and the associated equilibrium distribution. Our results are applied to optimal lowpass filters and matrix iterations.

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We compute approximate Fekete points for weighted polynomial interpolation, by a recent algorithm based on QR factorizations of Vandermonde matrices. We consider in particular the case of univariate and bivariate functions with prescribed poles or other singularities, which are absorbed in the basis by a weight function. Moreover, we apply the method to the construction of real and complex weighted polynomial filters, where the relevant concept is that of weighted norm.

the interactive creation and visualization of conformal maps to regions bounded by polygons. The most recent release supports new features, including an object-oriented command-line interface model, new algorithms for multiply elongated and multiple-sheeted regions, and a module for solving Laplace’s equation on a polygon with Dirichlet and homogeneous Neumann conditions. Brief examples are given to demonstrate the new capabilities.