Cubature formulas on a sphere that are invariant with respect to a group of dihedron rotations with inversion D 6h

Abstract

An algorithm of searching for the best (in a sense) cubature formulas on a sphere that are invariant with respect to a group of dihedron rotations with inversion D6h is developed. This algorithm is applied to find parameters of all the best cubature formulas of this group of symmetry up to the 23rd order of accuracy n. In the course of the study performed, exact values of parameters of the corresponding cubature formulas are found for n ≤ 11, and approximate values are obtained by numerical solving systems of nonlinear algebraic equations by a Newton-type method for other values of n. For the first time, ways of obtaining the best cubature formulas for a sphere are systematically investigated for the case of a group that is not a subgroup of the groups of symmetry of regular polyhedrons.