Numerical integration over the -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 . Table I lists orthogonal polynomials, coordinates and coefficients for integration points in the angular rules for 3rd and 7th degree precision and for . 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 .

**[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.**[4]**H. Fishman, ``Numerical integration constants,''*MTAC*, v. 11, 1957, p. 1-9. MR**0086391 (19:177g)****[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