Symmetric Quadrature Formulae for Simplexes

Symmetric interpolation polynomials are defined for N-dimensional simplexes with the aid of a symmetric coordinate notation. These polynomials are used to produce symmetric interpolatory quadrature formulae of arbitrary degree of precision over simplexes of arbitrary dimensionality. Tabulated values of weight coefficients are given for triangles and tetrahedra. 1. Introduction. Interpolative quadrature formulae for N-dimensional simplexes have been given by various authors, e.g., Stroud (1) or Hammer and Stroud (2). Their principal attraction in applications lies in the fact that multidimensional regions of integration can often be closely approximated by unions of simplexes. The object of this paper is to show that, for any N, it is possible to define quadrature formulae of any degree of precision n, symmetric in the sense of Hammer, Marlowe, and Stroud (3); and to give a straightforward procedure for finding the weights and node locations. Although the resulting quadrature formulae are not efficient in the sense of (3), they possess the advantage of being very convenient computationally, and can be generated easily for any reasonable values of N and n. They represent a natural generalization of the Newton-Cotes formulae to the N-dimensional case, and include the latter for N = 1.