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

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