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Asymptotic Expansions for Quadrature Errors over a Simplex

  • A. van der Sluis
Chapter
Part of the ISNM 57: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique book series (ISNM, volume 57)

Abstract

In [3] and [4] J.N. Lyness proved that the error functionals of repeated quadrature rules on a simplex have asymptotic expansions, in which, under circumstances, the odd terms vanish. The proof, based on Bernoulli functions and higher dimensional Euler-McLaurin expansions, is rather intricate.

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References

  1. [1]
    Edwards, R.E.: Functional analysis; theory and applications. Holt, Rinehart and Winston, New York, etc.: 1965.Google Scholar
  2. [2]
    Lyness, J.N.: Symmetric integration rules for hypercubes. Math. Comp. 19(1965), pp. 260–276.Google Scholar
  3. [3]
    Lyness, J.N.: Quadrature over a simplex: Part 1. A representation for the integrand function. SIAM J. Numer. Anal. 15(1978), pp. 122–133.CrossRefGoogle Scholar
  4. [4]
    Lyness, J.N.: Quadrature over a simplex: Part 2. A representation for the error functional. SIAM J. Numer. Anal. 15(1978), pp. 870–887.CrossRefGoogle Scholar
  5. [5]
    Lyness, J.N., and Genz, A.C.: On simplex trapezoidal rule families. SIAM J. Numer. Anal. 17(1980), pp. 126–147.CrossRefGoogle Scholar
  6. [6]
    Sluis, A. van der: The remainder term in quadrature formulae. Numer. Math. 19(1972), pp. 49–55.CrossRefGoogle Scholar
  7. [7]
    Whitney, H.: Analytic extensions of differentiable functions on closed sets. Trans. Amer. Math. Soc. 36(1934), pp. 63–89.CrossRefGoogle Scholar

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© Springer Basel AG 1982

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  • A. van der Sluis

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