Quadrature formulas based on rational interpolation

We consider quadrature formulas based on interpolation using the basis functions $1/(1+t_kx)$ $(k=1,2,3,\ldots)$ on $[-1,1]$, where $t_k$ are parameters on the interval $(-1,1)$. We investigate two types of quadratures: quadrature formulas of maximum accuracy which correctly integrate as many basis functions as possible (Gaussian quadrature), and quadrature formulas whose nodes are the zeros of the orthogonal functions obtained by orthogonalizing the system of basis functions (orthogonal quadrature). We show that both approaches involve orthogonal polynomials with modified (or varying) weights which depend on the number of quadrature nodes. The asymptotic distribution of the nodes is obtained as well as various interlacing properties and monotonicity results for the nodes

Harmonic Shears and Numerical Conformal Mappings

In this article we introduce a numerical algorithm for finding harmonic mappings by using the shear construction introduced by Clunie and Sheil-Small in 1984. The MATLAB implementation of the algorithm is based on the numerical conformal mapping package, the Schwarz-Christoffel toolbox, by T. Driscoll. Several numerical examples are given. In addition, we discuss briefly the minimal surfaces associated with harmonic mappings and give a numerical example of minimal surfaces.Comment: 15 pages, 6 figure