The Euler-Maclaurin expansion for the simplex

Abstract:

A natural extension of the one-dimensional trapezoidal rule to the simplex $0 \leqq {x_i} \leqq 1, \sum {x_i} \leqq 1$, is a rule Rf which uses as abscissas all those points on a hyper-rectangular lattice of spacing $h = 1/m$ which lie within the simplex, assigning an equal weight to each interior point. In this paper, rules of this type are defined and some of their properties are derived. In particular, it is shown that the error functional satisfies an Euler-Maclaurin expansion of the type \[ Rf - If \sim {A_1}h + {A_2}{h^2} + \cdots + {A_p}{h^p} + O({h^{p + 1}})\] so long as $f({\text {x}})$ and its partial derivatives of order up to p are continuous. Conditions under which this asymptotic series terminates are given, together with the condition for odd terms to drop out leaving an even expansion. The application to Romberg integration is discussed.