On the Structure of Fully Symmetric Multidimensional Quadrature Rules
P. Keast, James N. Lyness · SIAM Journal on Numerical Analysis · 1979
One method for constructing fully symmetric quadrature rules of specified moderate polynomial degree for fully symmetric integration regions consists of solving directly the possibly large system of moment fitting equations which define the problem. A familiar hazard is to find these equations are inconsistent. In a recent fundamental paper, Mantel and Rabinowitz [4] (SIAM J. Numer. Anal., 1977) treated this problem in some detail in a three dimensional context. They defined and calculated a set of consistency conditions and using these, systematized to a significant extent the present state of the art for two and three dimensional fully symmetric quadrature rules. This paper is complementary to [4]. We introduce a set of null spaces. Using these, the calculations of the consistency conditions which is in general tedious, can be reduced to an automatic linear procedure in a way which makes human error significantly less likely. In addition, these spaces may be used in the actual organization of a calculation for constructing quadrature rules. The linear aspect of a large system of nonlinear equations may be factored out, and the large system may be replaced by a set of much smaller nonlinear systems which may be solved sequentially.