New Euler-Maclaurin expansions and their application to quadrature over the -dimensional simplex
Abstract: The -panel offset trapezodial rule for noninteger values of , is introduced in a one-dimensional context. An asymptotic series describing the error functional is derived. The values of 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.
-  Christopher T. H. Baker and Graham S. Hodgson, Asymptotic expansions for integration formulas in one or more dimensions, SIAM J. Numer. Anal. 8 (1971), 473–480. MR 0285115, https://doi.org/10.1137/0708043
-  Axel Grundmann and H. M. Möller, Invariant integration formulas for the 𝑛-simplex by combinatorial methods, SIAM J. Numer. Anal. 15 (1978), no. 2, 282–290. MR 488881, https://doi.org/10.1137/0715019
-  J. N. Lyness, Quadrature over a simplex. I. A representation for the integrand function, SIAM J. Numer. Anal. 15 (1978), no. 1, 122–133. MR 0468118, https://doi.org/10.1137/0715008
-  J. N. Lyness and K. K. Puri, The Euler-Maclaurin expansion for the simplex, Math. Comp. 27 (1973), 273–293. MR 0375752, https://doi.org/10.1090/S0025-5718-1973-0375752-1
-  A. H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. Prentice-Hall Series in Automatic Computation. MR 0327006
- C. T. H. BAKER & G. S. HODGSON, "Asymptotic expansions for integration formulas in one or more dimensions," SIAM J. Numer. Anal., v. 8, 1971, pp. 473-480. MR 0285115 (44:2339)
- A. GRUNDMANN & H. M. MÖLLER, "Invariant integration formulas for the n-simplex by combinatorial methods," SIAM J. Numer. Anal., v. IS, 1978, pp. 282-290. MR 488881 (81e:41045)
- J. N. LYNESS, "Quadrature over a simplex: Part 1. A representation of the integrand function," SIAM J. Numer. Anal., v. 15, 1978, pp. 122-133. MR 0468118 (57:7957)
- J. N. LYNESS & K. K. PURI, "The Euler-Maclaurin expansion for the simplex," Math. Comp., v. 27, 1973, pp. 273-293. MR 0375752 (51:11942)
- A. H. STROUD, Approximate Calculation of Multiple Integrals, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR 0327006 (48:5348)