Symmetric integration rules for hypercubes. II. Rule projection and rule extension
Author:
J. N. Lyness
Journal:
Math. Comp. 19 (1965), 394407
MSC:
Primary 65.55
MathSciNet review:
0201068
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Abstract: A theory is described which facilitates the construction of highdimensional integration rules. It is found that, for large , an dimensional integration rule of degree man be constructed requiring a number of function evaluations of order . In an example we construct a dimensional rule of degree 9 which requires 52,701 function evaluations. The corresponding number for the product Gaussian is .
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(34 #952), http://dx.doi.org/10.1090/S00255718196502010673
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 C. F. Gauss, ``Methodus nova integralium valores per approximationem inveniendi,'' Comm. Soc. Reg. Sci. Göltingen, 1816. ,
 [2]
 P. C. Hammer, O. J. Marlowe & A. H. Stroud, ``Numerical integration over Simplexes and cones,'' MTAC, v. 10, 1956, pp. 130137. MR 19, 177. MR 0086389 (19:177e)
 [3]
 P. C. Hammer & A. H. Stroud, ``Numerical evaluation of multiple integrals. II,'' MTAC, v. 12, 1958, pp. 272280. MR 21 #970. MR 0102176 (21:970)
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 Z. Kopal, ``Numerical analysis,'' Wiley, New York and Chapman and Hall, London 1955. MR 17, 1007. MR 0077213 (17:1007c)
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 J. N. Lyness, ``Symmetric integration rules for hypercubes. I. Error coefficients,'' Math. Comp., v. 19, 1965, pp. 260276. MR 0201067 (34:952)
 [6]
 J. C. P. Miller, ``Numerical quadrature over a rectangular domain in two or more dimensions. Quadrature in several dimensions using special points,'' Math. Comp., v. 14, 1960, pp. 130138. MR 22 #6075.
 [7]
 F. Stenger, ``Numerical Integration in n Dimensions,'' Thesis, University of Alberta, Canada, 1963.
 [8]
 A. H. Stroud, ``Numerical integration formulas of degree 3 for product regions and cones,'' Math. Comp., v. 15, 1961, pp. 143150. MR 22 #12717. MR 0121990 (22:12717)
 [9]
 H. C. Thacher, Jr., ``Optimum quadrature formulas in dimensions,'' MTAC, v. 11, 1957, pp. 189194.
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DOI:
http://dx.doi.org/10.1090/S00255718196502010685
PII:
S 00255718(1965)02010685
Article copyright:
© Copyright 1965
American Mathematical Society
