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Numerical integration over the $ n$-dimensional spherical shell


Author: D. Mustard
Journal: Math. Comp. 18 (1964), 578-589
MSC: Primary 65.55
DOI: https://doi.org/10.1090/S0025-5718-1964-0170474-9
MathSciNet review: 0170474
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Abstract: The n-dimensional generalisation of a theorem by W. H. Peirce [1] is given, providing a method for constructing product type integration rules of arbitrarily high polynomial precision over a hyperspherical shell region and using a weight function $ {r^s}$. Table I lists orthogonal polynomials, coordinates and coefficients for integration points in the angular rules for 3rd and 7th degree precision and for $ n = 3(1)8$. Table II gives the radial rules for a shell of internal radius R and outer radius 1: (i) a formula for the coordinate and coefficient in the 3rd degree rule for arbitrary n, R; (ii) a formula for the coordinates and coefficients for the 7th degree rule for arbitrary n and R = 0 and (iii) a table of polynomials, coordinates and coefficients to 9D for n = 4, 5 and $ R = 0,\tfrac{1}{4},\tfrac{1}{2},\tfrac{3}{4}$.


References [Enhancements On Off] (What's this?)

  • [1] W. H. Peirce, ``Numerical integration over the spherical shell,'' MTAC, v. 11, 1957, p. 244-249. MR 0093910 (20:430)
  • [2] Z. Kopal, Numerical Analysis, Wiley, New York, 1955, p. 367-386. MR 0077213 (17:1007c)
  • [3] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. Vol. 23, Amer. Math. Soc., Providence, R. I., 1939.
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  • [5] A. H. Stroud & D. Secrest, ``Approximate integration formulas for certain spherically symmetric regions,'' Math. Comp., v. 17, 1963, p. 105-135. MR 0161473 (28:4677)
  • [6] I. P. Mysovskih, ``Cubature formulas for evaluating integrals over a sphere,'' Dokl. Akad. Nauk SSSR, v. 147, No. 3, 1962, p. 552-555. (Russian) MR 0146961 (26:4480)

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DOI: https://doi.org/10.1090/S0025-5718-1964-0170474-9
Article copyright: © Copyright 1964 American Mathematical Society

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