Advertisement

Construction of Known and New Cubature Formulas of Degree Five for Three-Dimensional Symmetric Regions, Using Orthogonal Polynomials

  • Ann Haegemans
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

We are concerned with determining the nodes and weights in a cubature formula of the form
$$\iiint\limits_R {w(x,y,z)f(x,y,z)}dx{\kern 1pt} dy{\kern 1pt} dz \simeq \sum\limits_{i = 1}^N {{w_i}f({x_i},{y_i},{z_i})} $$
(1)
which is exact for all polynomials in x, y and z of degree ≤ 5 but not for all polynomials of degree 6. R is a region in the three-dimensional Euclidian space, assumed to be symmetric with respect to the three axes. The weight function w(x, y, z) will be assumed to be symmetric in x, y and z: w(x, y, z) = w(‒x, y, z) = w(x,‒y, z) = w(x, y,‒z) ≤ 0

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Brown, W.S. Altran user’s manual. Murray Hill, New Jersey, Bell Laboratories 1973.Google Scholar
  2. [2]
    Ditkin, V.A. On certain approximate formulas for the calculation of triple integrals (in Russian), Dokl. Akad. Nauk. SSSR, 62, 1948, pp. 445–447.Google Scholar
  3. [3]
    Hammer, P.C., and Stroud, A.H. Numerical evaluation of multiple integrals II, Math. Tables Aids Comput., 12, 1958, pp. 272–280.CrossRefGoogle Scholar
  4. [4]
    Mustard, D., Lyness, J.N., and Blatt, J.M. Numerical quadrature in N dimensions, Computer J., 6, 1963–1964, pp. 75–87.CrossRefGoogle Scholar
  5. [5]
    Sadowsky, M. A formula for the approximate computation of a triple integral, Amer. Math. Monthly, 47, 1940, pp. 539–543.CrossRefGoogle Scholar
  6. [6]
    Stroud, A.H. Some fifth degree integration formulas for symmetric regions II, Numer. Math., 9, 1967, pp. 460–468.CrossRefGoogle Scholar
  7. [7]
    Stroud, A.H. Approximate calculation of multiole integrals, Englewood Cliffs, N.J. Prentice Hall, 1971.Google Scholar
  8. [8]
    Stroud, A.H., and Secrest, D. Approximate integration formulas for certain spherically symmetric regions, Math. Comput., 17, 1963, pp. 105–135.CrossRefGoogle Scholar
  9. [9]
    Tyler, G.W. Numerical integration of functions of several variables, Canad. J. Math., 5, 1953, pp. 393–412.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1982

Authors and Affiliations

  • Ann Haegemans

There are no affiliations available

Personalised recommendations