On Self-Contained Numerical Integration Formulas for Symmetric Regions

Abstract

Approximate integration formulas for compact regions in n-dimensional Euclidean space are considered. For planar regions which are invariant under the symmetries of a square, it is shown that there exist 4-point equally-weighted formulas of (algebraic) degree three with all of the points in the domain of integration. However, for $n \geqq 3$ and odd, there exist regions which have the symmetries of an n-cube, such that none of the $2n$-point formulas of degree three are self-contained. An example is given of such a region. For planar regions which are invariant under the symmetries of a triangle, it is shown that there exist 3-point equally-weighted formulas of degree two with all points in the domain of integration. It is again shown by example that this property does not extend to $n=3$. The planar results are the first to guarantee self-contained formulas with the minimal number of points for reasonably large classes of regions.