Generating Gauss quadratures for Green's function /sup 1///sub r/: a randomized algorithm

Abstract

We report a randomized scheme for generating Gauss quadratures for an exponential integral representation of the Green's function /sup 1///sub r/. These Gauss quadratures form the basis of the exponential-expansion-based method, which has previously been developed for rapid and accurate evaluation of the potential field and its gradient in three dimensions. Given a desired degree of accuracy on the approximation of /sup 1///sub r/, the technique proposed here enables generation of exponential expansion with sizes as small as possible. It makes use of the standard Legendre-Gauss and Chebychev-Gauss quadratures, and does not require solving a large system of non-linear equations.