Quadrature over a simplex. II. A representation for the error functional

A particular subdivision of a simplex S/sub n/ into m/sup n/ subsimplices of equal n-volume is described. The m/sup n/-copy quadrature rule Q/sup (m)/(S/sub n/) is defined in terms of a set (Q) of n-factorial simplex quadrature rules, one for each distinct type of subsimplex occurring in the subdivision. An n-dimensional analog of the Euler--Maclaurin asymptotic expansion for the simplex valid for analytic integrand functions phi and any quadrature rule set (Q) is derived by integrating the representation for phi(x) obtained in Part 1. This has the form Q/sup (m)/(S/sub n/)phi - I(S/sub n/)phi asympotically equals ..sigma.. A/sub q/(Q; phi)/m/sup q/, and terminates when phi(x) is a polynomial. For rule sets based on many familiar simplex rules, the odd terms vanish to leave an even expansion.