Constructing Cubature Formulas of Degree 5 with Few Points

TL;DR

This paper develops efficient fifth-degree cubature formulas for n-cubes and spherically symmetric regions, minimizing the number of points needed for accurate numerical integration in multiple dimensions.

Contribution

It introduces new cubature formulas with fewer points than existing methods, achieving minimal points for certain dimensions.

Findings

For n-cubes, formulas use at most n^2+5n+3 points.

For spherically symmetric regions, formulas use at most n^2+3n+3 points.

Minimal point formulas are achieved when n=7, with n^2+n+1 points.

Abstract

This paper will devote to construct a family of fifth degree cubature formulae for $n$-cube with symmetric measure and $n$-dimensional spherically symmetrical region. The formula for $n$-cube contains at most $n^2+5n+3$ points and for $n$-dimensional spherically symmetrical region contains only $n^2+3n+3$ points. Moreover, the numbers can be reduced to $n^2+3n+1$ and $n^2+n+1$ if $n=7$ respectively, the later of which is minimal.