67 new formulas were found for multi-dimensional integration over standard regions, including the sphere and Gaussian weighted integrals over all space, using fewer integration points than known formulas of the same degree.
67 new cubature rules are found for three standard multi-dimensional integrals with spherically symmetric regions and weight functions using direct search with a numerical zero-finder. 63 of the new rules have fewer integration points than known rules of the same degree, and 20 are within three points of Möller’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 the integration over the sphere and Gaussian-weighted integrals over the entire space, each in 6 and 7 dimensions, with 127 and 183 points, respectively.
This paper originally appeared in Electronic Transactions on Numerical Analysis, Volume 51, pp. 219–239, 2019. DOI: 10.1553/etna_vol51s219
NOTE: Several of the MATLAB/Octave and Maxima programs used to find these rules and to refine them to high precision are available in a zip file from the publisher.