Centroid Method of Integration

Abstract

THE usefulness, for numerical integration in p dimensions, of the following “centroid method” seems to have been generally underestimated. The method is a generalization of the mid-point method for one dimension. The first approximation to the integral is f(x̄) multiplied by the “volume” of the region of integration, where x̄ is the centroid of the region. This first approximation is itself often more accurate than might be expected. (It is exact if f is linear.) If f has a p-dimensional Taylor expansion about the centroid, valid in the region, then the integral is given exactly by the following equation where n = ν₁ + ν₂ + … + ν_p, where the terms for n = 1 vanish, and where m(ν) is the generalized moment of inertia Because any non-pathological domain can be “triangulated” into simplexes, it is convenient to have a formula for the generalized moments of inertia of a simplex the vertices of which are x_k = (x_k1, …, x_kp), (k = 0,1, …, p). It can be shown that m(ν) is equal to the coefficient of ${t_1}^{{\text{ν}}_1}{t_2}^{{\text{ν}}_2}$ … in where V is the “volume” of the simplex, given by a familiar determinant, and where If 1≤n≤3 (but not if n≥4), it can be readily deduced that and if p = 2 and 1≤n≤5 (but not if n≥6), where c₂=1/12, c₃ = c₄=1/30, c₅ = 2/105. An example of a fourth moment for arbitrary p is