Cubature Formulas for Symmetric Measures in Higher Dimensions with Few Points

Details Cubature Formulas for Symmetric Measures in Higher Dimensions with Few Points Cubature Formulas for Symmetric Measures in Higher Dimensions with Few Points

2005-09-05

arxiv.org/abs/math/0509101v2

Mathematics

Numerical Analysis

19 pages, Mathematics of Computation, to appear

Scientific paper

We study cubature formulas for d-dimensional integrals with an arbitrary symmetric weight function of tensor product form. We present a construction that yields a high polynomial exactness: for fixed degree l=5 or l=7 and large dimension, the number of knots is only slightly larger than the lower bound of M\"oller and much smaller compared to the known constructions. We also show, for any odd degree l=2k+1, that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.

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