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

Author:
Elise de Doncker

Journal:
Math. Comp. **33** (1979), 1003-1018

MSC:
Primary 65B15; Secondary 65D32

MathSciNet review:
528053

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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.

**[1]**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****[2]**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**, 10.1137/0715019**[3]**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****[4]**J. N. Lyness and K. K. Puri,*The Euler-Maclaurin expansion for the simplex*, Math. Comp.**27**(1973), 273–293. MR**0375752**, 10.1090/S0025-5718-1973-0375752-1**[5]**A. H. Stroud,*Approximate calculation of multiple integrals*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. Prentice-Hall Series in Automatic Computation. MR**0327006**

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DOI:
https://doi.org/10.1090/S0025-5718-1979-0528053-8

Article copyright:
© Copyright 1979
American Mathematical Society