Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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

Author: Elise de Doncker
Journal: Math. Comp. 33 (1979), 1003-1018
MSC: Primary 65B15; Secondary 65D32
MathSciNet review: 528053
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [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)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65B15, 65D32

Retrieve articles in all journals with MSC: 65B15, 65D32

Additional Information

Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society