On the Existence of Regions with Minimal Third Degree
نویسنده
چکیده
A. H. Stroud has shown that n + 1 is the minimum possible number of nodes in an integration formula of degree three for any region in E„. In this paper, in answer to the question of the attainability of this minimal number, we exhibit for each n a region that possesses a third degree formula with n + 1 nodes. This is accomplished by first deriving an in + 2)-point formula of degree three for an arbitrary region that is invariant under the group of affine transformations that leave an «-simplex fixed. The formula is then applied to a one-parameter family of such regions, and a value of the parameter is determined for which the weight at the centroid vanishes.
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تاریخ انتشار 2010