Crossproduct cubature error bounds
Author:
Frank G. Lether
Journal:
Math. Comp. 24 (1970), 583592
MSC:
Primary 65.55
MathSciNet review:
0275673
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Abstract: This paper is concerned with crossproduct cubature rules. We use Sard's Kernel Theorem to express the crossproduct cubature error in terms of one variable kernels. This simplified representation of the error is then used to derive cubature error bounds analogous to those obtained by Secrest and Stroud , for quadrature rules.
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 A. C. Ahlin, "On error bounds for Gaussian cubature," SIAM Rev., v. 4, 1962, pp. 2539. MR 27 #2105. MR 0152125 (27:2105)
 [2]
 R. E. Barnhill, "An error analysis for numerical multiple integration. I," Math. Comp., v. 22, 1968, pp. 98109. MR 37 #2438. MR 0226852 (37:2438)
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 M. M. Chawla, "On the estimation of errors of Gaussian cubature formulas," SIAM J. Numer. Anal., v. 5, 1968, pp. 172181. MR 36 #7324. MR 0224280 (36:7324)
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 P. I. Davis, "Errors of numerical approximation for analytic functions," J. Rational Mech. Anal., v. 2, 1953, pp. 303313. MR 14, 907. MR 0054348 (14:907f)
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 P. J. Davis, Interpolation and Approximation, Blaisdell, Waltham, Mass., 1963. MR 28 #393. MR 0157156 (28:393)
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 I. A. Èzrohi, "General forms of remainder terms of linear formulae in multidimensional approximate analysis. I, II," Mat. Sb., v. 38(80), 1956, pp. 389416; v. 43(85), 1957, pp. 928. (Russian) MR 18, 32; MR 19, 1199.
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 F. G. Lether, CrossProduct Cubature Error Estimates, Ph.D. Thesis, The University of Utah, Salt Lake City, Utah, 1969.
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 J. N. Lyness & B. J. J. McHugh, "Integration over multidimensional hypercubes. I: A progressive procedure," Comput. J., v. 6, 1963, pp. 264270.
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 S. M. Nikol'skiĭ, Quadrature Formulas, Fizmatgiz, Moscow, 1958. (Russian)
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 A. Sard, Linear Approximation, Math. Surveys, no. 9, Amer. Math. Soc., Providence, R.I., 1963. MR 0158203 (28:1429)
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 D. D. Stancu, "The remainder of certain linear approximation formulas in two variables," J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., v. 1, 1964, pp. 137163. MR 31 #1503. MR 0177240 (31:1503)
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 F. Stenger, "Error bounds for the evaluation of integrals by repeated Gausstype formulae," Numer. Math., v. 9, 1966, pp. 200213, MR 34 #5289. MR 0205462 (34:5289)
 [13]
 A. H. Stroud & D. Secrest, Gaussian Quadrature Formulas, PrenticeHall, Englewood Cliffs, N. J., 1966. MR 34 #2185. MR 0202312 (34:2185)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197002756736
PII:
S 00255718(1970)02756736
Keywords:
Sard kernels,
Gaussian rules,
Peano kernels,
crossproduct rules,
cubature error bounds
Article copyright:
© Copyright 1970
American Mathematical Society
