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The Computer Journal 1963 6(1):75-87; doi:10.1093/comjnl/6.1.75
© 1963 by British Computer Society
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Numerical quadrature in n dimensions

D. Mustard, J. N. Lyness and J. M. Blatt

University of New South Wales, Box 1, post Office, Kensington, N.S.W., Australia

We investigate a selection of integration rules based on the combination of third degree rules for elementary three-dimensional sub-domains. The practical problems associated with the application of these rules to domains bounded by planes of a certain type are discussed in detail. These included integration rules for less symmetrical domains which may occur near the boundary of the volume of integration, methods for combining integration coefficients from adjacent sub-domains, and methods for changing net size within the volume of integration.

Particular attention is paid to minimizing the number of points at which the function has to be evaluated, and error estimates in terms of computation time are given. A list of integration coefficients of general interest for three-dimensional integrations is presented. The discussion is generalized to n dimensions for hyper-cubic domains.


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