Partially Symmetric Cubature Formulas for Even Degrees of Exactness

Abstract

The method presented consists of a reduction of the moment equation system and an a priori assumption of the structure of a formula. It allows the determination of partially symmetric cubature formulas for regions with the same or higher symmetries. This makes it specially suited for even degrees of exactness, because these cubature formulas exhibit only partial symmetry. The size of the resulting nonlinear system of equations calls for a numerical solution. Several new results are given for squares, circles, and hexagons. The application to higher-dimensional problems is straightforward. However, no systematic search has been conducted as yet.