Numerical integration over the -dimensional spherical shell

Author:
D. Mustard

Journal:
Math. Comp. **18** (1964), 578-589

MSC:
Primary 65.55

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]**William H. Peirce,*Numerical integration over the spherical shell*, Math. Tables Aids Comput**11**(1957), 244–249. MR**0093910**, 10.1090/S0025-5718-1957-0093910-1**[2]**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****[3]**G. Szegö,*Orthogonal Polynomials*, Amer. Math. Soc. Colloq. Publ. Vol. 23, Amer. Math. Soc., Providence, R. I., 1939.**[4]**Herbert Fishman,*Numerical integration constants*, Math. Tables Aids Comput.**11**(1957), 1–9. MR**0086391**, 10.1090/S0025-5718-1957-0086391-5**[5]**A. H. Stroud and Don Secrest,*Approximate integration formulas for certain spherically symmetric regions*, Math. Comp.**17**(1963), 105–135. MR**0161473**, 10.1090/S0025-5718-1963-0161473-0**[6]**I. P. Mysovskih,*Cubature formulas for evaluating integrals over a sphere*, Dokl. Akad. Nauk SSSR**147**(1962), 552–555 (Russian). MR**0146961**

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DOI:
https://doi.org/10.1090/S0025-5718-1964-0170474-9

Article copyright:
© Copyright 1964
American Mathematical Society