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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

Numerical integration over the $ n$-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 $ {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}$.


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