Efficient Cubature Rules
arXiv:1712.07309 · doi:10.1553/etna_vol51s219
Abstract
73 new cubature rules are found for three standard multidimensional integrals with spherically symmetric regions and weights, using direct search with a numerical zero-finder. All but four of the new rules have fewer integration points than known rules of the same degree, and twenty are within three points of M{ö}ller's lower bound. Most have all positive coefficients and most have some symmetry, including some supported by one or two concentric spheres. They include degree 7 formulas for integration over the sphere and Gaussian-weighted integrals over all space, each in 6 and 7 dimensions, with 127 and 183 points, respectively.
26 pages, 4 figures, 95 supplemental files. The 10 point rule for E_2^{r^2} was known, but had not been published. It was found by Jankewitz (private communication to R. Cools, 1998). Both the 15 and 16 point formulas of degree 4 for the sphere in 4 dimensions are new, and different from the 16 point formula found by Mysovskih