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

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

MathSciNet review:
528053

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Abstract | References | Similar Articles | Additional Information

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]**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)****[2]**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)****[3]**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)****[4]**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)****[5]**A. H. STROUD,*Approximate Calculation of Multiple Integrals*, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR**0327006 (48:5348)**

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

Article copyright:
© Copyright 1979
American Mathematical Society