Generating Gauss quadratures for Green´s function 1/r: a randomized algorithm

We report a randomized scheme for generating Gauss quadratures for an exponential integral representation of the Green´s function ¹ /_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 ¹/_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