Numerical integration over the -dimensional spherical shell
Abstract: The n-dimensional generalisation of a theorem by W. H. Peirce  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 .
-  William H. Peirce, Numerical integration over the spherical shell, Math. Tables Aids Comput 11 (1957), 244–249. MR 0093910, https://doi.org/10.1090/S0025-5718-1957-0093910-1
-  Zdeněk Kopal, Numerical analysis. With emphasis on the application of numerical techniques to problems of infinitesimal calculus in single variable, John Wiley & Sons, Inc., New York, 1955. MR 0077213
-  G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. Vol. 23, Amer. Math. Soc., Providence, R. I., 1939.
-  Herbert Fishman, Numerical integration constants, Math. Tables Aids Comput. 11 (1957), 1–9. MR 0086391, https://doi.org/10.1090/S0025-5718-1957-0086391-5
-  A. H. Stroud and Don Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (1963), 105–135. MR 0161473, https://doi.org/10.1090/S0025-5718-1963-0161473-0
-  I. P. Mysovskih, Cubature formulas for evaluating integrals over a sphere, Dokl. Akad. Nauk SSSR 147 (1962), 552–555 (Russian). MR 0146961
- W. H. Peirce, ``Numerical integration over the spherical shell,'' MTAC, v. 11, 1957, p. 244-249. MR 0093910 (20:430)
- Z. Kopal, Numerical Analysis, Wiley, New York, 1955, p. 367-386. MR 0077213 (17:1007c)
- G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. Vol. 23, Amer. Math. Soc., Providence, R. I., 1939.
- H. Fishman, ``Numerical integration constants,'' MTAC, v. 11, 1957, p. 1-9. MR 0086391 (19:177g)
- 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)
- 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)
Retrieve articles in Mathematics of Computation with MSC: 65.55
Retrieve articles in all journals with MSC: 65.55