Abstract
For multidimensional integration, there exist some well-known families of fully-symmetric rules of different degrees and dimensions, for example, Phillips (1967), Stenger (1971), Keast (1979), Genz and Malik (1980, 1983). In two and three dimensions, there also exist several rules of different degrees and over different regions. The only rules for the 4-cube we have been able to find are one degree 7 rule by Stroud (1967) and one degree 5 rule by Stroud and Goit (1968). In this talk we will discuss efficient rules of degrees 7 and 9.
The main tools we are uising are the consistency conditions for rules to be of a particular degree developed by Keast and Lyness (1979). We are mainly interested in good rules (rules with all the evaluation points inside the 4-cube, and all the weights positivw). In a recent paper, Espelied (1984) has shown how to use the theory of moments for polynomials in constructing good fully-symmetric rules (FSG-rules) and has used technique in 3 dimensions. We will use the technique in 4 dimensions. Using this technique we were able to construct the minimum point, FSG-rules for all degrees up to and including degree 9. We were also bale to construct pairs of rules of degrees 9 and 7 and of degrees 7 and 5. We will also present on some tests on some of the new rules of degrees 9 to 7 and compare them with ADAPT and the degree 9 version of Genz and Malik’s “embedded family”.
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References
T.O. Espelid, On the construction of good fully symmetric integration rules, Report in informatics, No. 13, (1984), Dept. of Informatics, University of Bergen, Norway.
A.C. Genz and A.A. Malik, Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region, J. Comput. Appl. Math., 6 (1980), pp. 295–302.
A.C. Genz and A.A. Malik, An imbedded family of fully symmetric numerical integration rules, SIAM J. Numer. Anal., 20 (1983), pp. 580–588.
P. Keast, Some fully symmetric quadrature formulae for product spaces, J. Inst. Maths. Applies. 23 (1979), pp. 251–264.
P. Keast and J.N. Lyness, On the structure of fully-symmetric multidimensional quadrature rules, SIAM J. Numer. Anal., 16 (1979), pp. 11–29.
G.M. Phillips, Numerical integration over an N-dimensional rectangular region, Comput. J., 10 (1967), pp. 297–299.
F. Stenger, Tabulation of certain fully symmetric formulae of degree 7, 9, and 11, Math. Comp., 25 (1971), Microfiche Supplement.
A.H. Stroud, Some seventh degree integration formulas for symmetric region, SIAM J. Numer. Anal., 4 (1967), pp. 37–44.
A.H. Stroud and E.H. Goit, Jr., Some extensions of integration formulas, SIAM J. Numer. Anal., 5 (1968), pp. 243–251.
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© 1987 D. Reidel Publishing Company
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Sorevik, T., Espelid, T.O. (1987). Fully Symmetric Integration Rules for the Unit Four-Cube. In: Keast, P., Fairweather, G. (eds) Numerical Integration. NATO ASI Series, vol 203. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3889-2_18
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DOI: https://doi.org/10.1007/978-94-009-3889-2_18
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