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 is the minimum possible number of nodes in an integration formula of degree three for any region in . In this paper, in answer to the question of the attainability of this minimal number, we exhibit for each a region that possesses a third degree formula with nodes. This is accomplished by first deriving an -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.

**[1]**R. J. De Vogelaere, Private communication.**[2]**F. N. Fritsch,*On Minimal Positive and Self-Contained Multi-Dimensional Integration Formulas*, Ph.D. Thesis, University of California, Berkeley, Calif., 1969. (Available as UCRL-50600, Lawrence Radiation Laboratory, Livermore, Calif.)**[3]**Preston C. Hammer and Arthur H. Stroud,*Numerical integration over simplexes*, Math. Tables Aids Comput.**10**(1956), 137–139. MR**0086390**, https://doi.org/10.1090/S0025-5718-1956-0086390-2**[4]**Preston C. Hammer and A. Wayne Wymore,*Numerical evaluation of multiple integrals. I*, Math. Tables Aids Comput.**11**(1957), 59–67. MR**0087220**, https://doi.org/10.1090/S0025-5718-1957-0087220-6**[5]**I. P. Mysovskikh, "Proof of the minimality of the number of nodes in the cubature formula for a hypersphere,"*Ž. Vyčisl. Mat. i Mat. Fiz.*, v. 6, 1966, pp. 621-630 =*U.S.S.R. Comput. Math. and Math. Phys.*, v. 6, no. 4, 1966, pp. 15-27. MR**33**#8104.**[6]**Arthur H. Stroud,*Quadrature methods for functions of more than one variable*, Ann. New York Acad. Sci.**86**(1960), 776–791 (1960). MR**0119417****[7]**A. H. Stroud,*Numerical integration formulas of degree 3 for product regions and cones*, Math. Comp.**15**(1961), 143–150. MR**0121990**, https://doi.org/10.1090/S0025-5718-1961-0121990-4**[8]**A. H. Stroud,*A fifth degree integration formula for the 𝑛-simplex*, SIAM J. Numer. Anal.**6**(1969), 90–98. MR**0248981**, https://doi.org/10.1137/0706009

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Additional Information

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