A decomposition method to construct cubature formulae of degree 3

TL;DR

This paper introduces a decomposition method to construct third-degree cubature formulas in n-dimensional space using only 2n points, simplifying the process for symmetric integrals.

Contribution

It presents a novel decomposition approach that reduces the multidimensional problem to independent one-dimensional moment problems for degree 3 cubature formulas.

Findings

Constructed third-degree formulas with 2n points for symmetric integrals.

Simplified the multidimensional problem to n one-dimensional moment problems.

Provided a practical method for numerical integration in high dimensions.

Abstract

Numerical integration formulas in $n$-dimensional Euclidean space of degree three are discussed. For the integrals with permutation symmetry we present a method to construct its third-degree integration formulas with $2n$ real points. We present a decomposition method and only need to deal with $n$ one-dimensional moment problems independently.