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Constructing fully symmetric cubature formulae for the sphere


Authors: Sangwoo Heo and Yuan Xu
Journal: Math. Comp. 70 (2001), 269-279
MSC (2000): Primary 65D32, 41A55, 41A63
DOI: https://doi.org/10.1090/S0025-5718-00-01198-4
Published electronically: March 3, 2000
MathSciNet review: 1680883
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Abstract: We construct symmetric cubature formulae of degrees in the 13-39 range for the surface measure on the unit sphere. We exploit a recently published correspondence between cubature formulae on the sphere and on the triangle. Specifically, a fully symmetric cubature formula for the surface measure on the unit sphere corresponds to a symmetric cubature formula for the triangle with weight function $(u_{1}u_{2}u_{3})^{-1/2}$, where $u_{1}$, $u_{2}$, and $u_{3}$ are homogeneous coordinates.


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  • 1. Z. P. Bazant and B. H. Oh, Efficient numerical integration on the surface of a sphere, Z. Angew. Math. Mech. 66 (1986), 37-49. MR 87f:68024
  • 2. R. Cools and P. Rabinowitz, Monomial cubature rules since ``Stroud'': a compilation, J. Comp. Appl. Math. 48 (1993), 309-326. CMP 94:05
  • 3. D. A. Dunavant, High degree efficient symmetrical Gaussian quadrature rules for the triangle, Internat. J. Numer. Methods Engrg. 21 (1985), 1129-1148. MR 86h:65029
  • 4. H. Engels, Numerical quadrature and cubature, Academic Press, New York, 1980. MR 83g:65002
  • 5. S. Heo and Y. Xu, Constructing cubature formulae for spheres and balls, J. Comp. Appl. Math. 12 (1999), 95-119. CMP 2000:06.
  • 6. S. Heo and Y. Xu, Constructing cubature formulae for spheres and triangles, Technical Report, University of Oregon, 1998.
  • 7. P. Keast, Cubature formulas for the surface of the sphere, J. Comp. Appl. Math. 17 (1987), 151-172. MR 88e:65027
  • 8. P. Keast, Moderate-degree tetrahedral quadrature formulas, Comput. Methods Appl. Mech. Engrg. 55 (1986), 339-348. MR 87g:65035
  • 9. P. Keast and J. C. Diaz, Fully symmetric integration formulas for the surface of the sphere in $s$ dimensions, SIAM J. Numer. Anal. 20 (1983), 406-419. MR 84h:65025
  • 10. V.I. Lebedev, Quadrature on a sphere, USSR Comp. Math. and Math. Phys. 16 (1976), 10-24. MR 55:11578
  • 11. V.I. Lebedev, Spherical quadrature formulas exact to orders 25-29, Siberian Math. J. 18 (1977), 99-107. MR 56:7126
  • 12. V.I. Lebedev, A quadrature formula for the sphere of 59th algebraic order of accuracy, Russian Acad. Sci. Dokl. Math. 50 (1995), 283-286. CMP 95:06
  • 13. V.I. Lebedev and L. Skorokhodov, Quadrature formulas of orders 41,47 and 53 for the sphere, Russian Acad. Sci. Dokl. Math. 45 (1992), 587-592. MR 94b:65040
  • 14. J.N. Lyness and R. Cools, A survey of numerical cubature over triangles, Mathematics of Computation 1943-1993: a half-century of computational mathematics (Vancouver, BC, 1993), Proc. Sympos. Appl. Math., 48, Amer. Math. Soc., Providence, RI, 1994, pp. 127-150. MR 95j:65021
  • 15. J.N. Lyness and D. Jespersen, Moderate degree symmetric quadrature rules for the triangle, J. Inst. Maths. Applics. 15 (1975), 19-32. MR 51:14536
  • 16. J. I. Maeztu and E. Sainz de la Maza, Consistent structures of invariant quadrature rules for the $n$-simplex, Math. Comp. 64 (1995), 1171-1192. MR 95j:65022
  • 17. A. D. McLaren, Optimal numerical integration on a sphere, Math. Comp. 17 (1963), 361-383. MR 28:2635
  • 18. I. P. Mysovskikh, Interpolatory cubature formulas, ``Nauka'', Moscow, 1981 (Russian). MR 83i:65025
  • 19. S. L. Sobolev, Cubature formulas on the sphere invariant under finite groups of rotations, Sov. Math. Dokl. 3 (1962), 1307-1310. MR 25:4635
  • 20. A. Stroud, Approximate calculation of multiple integrals, Prentice Hall, Englewood Cliffs, NJ, 1971. MR 48:5348
  • 21. Y. Xu, On orthogonal polynomials in several variables, Special functions, $q$-series, and related topics, Fields Institute Communications, vol. 14, 1997, pp. 247 - 270. MR 99a:33009
  • 22. Y. Xu, Orthogonal polynomials and cubature formulae on spheres and on balls, SIAM J. Math. Anal. 29 (1998), 779-793. MR 99j:33015
  • 23. Y. Xu, Orthogonal polynomials and cubature formulae on spheres and on simplices, Methods Appl. Anal. 5 (1998), 169-184. CMP 98:16

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Additional Information

Sangwoo Heo
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
Email: yuan@math.uoregon.edu

Yuan Xu
Affiliation: Division of Science and Mathematics, University of Minnesota-Morris, Morris, Minnesota 56267
Address at time of publication: Department of Mathematics, University of Southern Indiana, Evansville, Indiana 47712
Email: sheo@cda.mrs.umn.edu

DOI: https://doi.org/10.1090/S0025-5718-00-01198-4
Keywords: Cubature formulae, on the unit sphere, on the triangle, symmetric formula on a triangle, octahedral symmetry
Received by editor(s): July 8, 1997
Received by editor(s) in revised form: February 6, 1998, July 14, 1998, and January 12, 1999
Published electronically: March 3, 2000
Additional Notes: Supported by the National Science Foundation under Grants DMS-9500532 and 9802265.
Article copyright: © Copyright 2000 American Mathematical Society

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