Abstract:
An algorithm of the search for the best (in a sense) cubature formulas on a sphere that are invariant with respect to the transformations of the icosahedral group of rotations with inversion is described. This algorithm is applied to finding the parameters of all the best cubature formulas of this symmetry type up to the $79$th order of accuracy. The parameters of the new cubature formulas of the $21$st, $25$th and $29$th orders of accuracy to $16$ significant digits are given.
Key words:numerical integration, invariant cubature formulas, invariant polynomials, icosahedral group of rotations.
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