Efficient integration on the hypersphere (Q1066612)

Various applications require the integration of functions on spherical surfaces in Euclidean 4-space. The concept of spherical t-designs confers an advantage for the solution of that class of numerical problems. By definition, all points of a t-design are equally weighted. Furthermore, suitably chosen spherical designs possess an automorphism group acting transitively on the points. The most interesting group in this context is \(I_ 4\), the hypericosahedral group of order 14,400. It allows the construction of a unique 19-design containing 3600 integration points furnishing the exact integration of hyperspherical harmonics up to 19th order, while 20th order harmonics are integrated with minimized errors.