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On the existence of regions with minimal third degree integration formulas


Author: F. N. Fritsch
Journal: Math. Comp. 24 (1970), 855-861
MSC: Primary 65.55
DOI: https://doi.org/10.1090/S0025-5718-1970-0277112-8
MathSciNet review: 0277112
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Abstract: 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_n}$. 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 $ (n + 2)$-point formula of degree three for an arbitrary region that is invariant under the group of affine transformations that leave an $ n$-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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DOI: https://doi.org/10.1090/S0025-5718-1970-0277112-8
Keywords: Approximate integration formulas, numerical integration formulas, minimal integration formulas, approximation of linear functionals, symmetric regions
Article copyright: © Copyright 1970 American Mathematical Society