Numerical integration formulas of degree two

Abstract

where R is a region in n-dimensional, real, euclidean space; x = (xi, X2, ** , ; the ai are constants; and the vP are points in the space. Most previous authors have given formulas for special regions (for a bibliography see [4]). Thacher [71 has given a method for constructing formulas of degree 2 with n + 1 points for general regions and of degree 3 with 2n points for certain symmetric regions; with his method, however, each region must also be treated separately. Our main results are to obtain specific formulas of degree 2 with n + 1 points for a general region satisfying a certain condition of non-degeneracy, and to show that for these regions such formulas cannot be obtained with fewer points. We also give a specific 2n point formula of degree 3 for a general centrally symmetric region. These results are a generalization of those of Georgiev [1, 2, 3] who has obtained similar results (but gives no general formulas) for n = 2, 3 with w(x) -1. Our results are obtained by a different method which was developed without knowledge of Georgiev's work.