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

  • Ann Haegemans
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)


We are concerned with determining the nodes and weights in a cubature formula of the form
$$\iiint\limits_R {w(x,y,z)f(x,y,z)}dx{\kern 1pt} dy{\kern 1pt} dz \simeq \sum\limits_{i = 1}^N {{w_i}f({x_i},{y_i},{z_i})} $$
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


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© Springer Basel AG 1982

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  • Ann Haegemans

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