Perfectly symmetric two-dimensional integration formulas with minimal numbers of points

Abstract

Perfectly symmetric integration formula of degrees 9–15 with a minimal number of points are computed for the square, the circle and the entire plane with weight functions exp ( − ( x 2 + y 2 ) ) ( - ({x^2} + {y^2})) and exp ( − ( x 2 + y 2 ) 1 / 2 ) ( - {({x^2} + {y^2})^{1/2}}) . These rules were computed by solving a large system of nonlinear algebraic equations having a special structure. In most cases where the minimal formula has a point exterior to the region or where some of the weights are negative, ’good’ formulas, which consist only of interior points and have only positive weights, are given which contain more than the minimal number of points.