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 | References | Similar Articles | Additional Information

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]**P. C. Hammer & A. H. Stroud, "Numerical integration over Simplexes,"*Math. Tables Aids Comput.*, v. 10, 1956, pp. 137-139. MR**19**, 177. MR**0086390 (19:177f)****[4]**P. C. Hammer & A. W. Wymore, "Numerical evaluation of multiple integrals. I,"*Math. Tables Aids Comput.*, v. 11, 1957, pp. 59-67. MR**19**, 323. MR**0087220 (19:323a)****[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]**A. H. Stroud, "Quadrature methods for functions of more than one variable,"*Ann. New York Acad. Sci.*, v. 86, 1960, pp. 776-791. MR**22**#10179. MR**0119417 (22:10179)****[7]**A. H. Stroud, "Numerical integration formulas of degree 3 for product regions and cones,"*Math. Comp.*, v. 15, 1961, pp. 143-150. MR**22**#12717. MR**0121990 (22:12717)****[8]**A. H. Stroud, "A fifth degree integration formula for the -simplex,"*SIAM J. Numer. Anal.*, v. 6, 1969, pp. 90-98. MR**0248981 (40:2231)**

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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