New Euler-Maclaurin expansions and their application to quadrature over the $s$-dimensional simplex

Abstract:

The $\mu$-panel offset trapezodial rule for noninteger values of $\mu$, is introduced in a one-dimensional context. An asymptotic series describing the error functional is derived. The values of $\mu$ for which this is an even Euler-Maclaurin expansion are determined, together with the conditions under which it terminates after a finite number of terms. This leads to a new variant of one-dimensional Romberg integration. The theory is then extended to quadrature over the s-dimensional simplex, the basic rules being obtained by an iterated use of one-dimensional rules. The application to Romberg integration is discussed, and it is shown how Romberg integration over the simplex has properties analogous to those for standard one-dimensional Romberg integration and Romberg integration over the hypercube. Using extrapolation, quadrature rules for the s-simplex can be generated, and a set of formulas can be obtained which are the optimum so far discovered in the sense of requiring fewest function values to obtain a specific polynomial degree.