Cross-product cubature error bounds

Author:
Frank G. Lether

Journal:
Math. Comp. **24** (1970), 583-592

MSC:
Primary 65.55

DOI:
https://doi.org/10.1090/S0025-5718-1970-0275673-6

MathSciNet review:
0275673

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with cross-product cubature rules. We use Sard's Kernel Theorem to express the cross-product 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.

**[1]**A. C. Ahlin,*On error bounds for Gaussian cubature*, SIAM Rev.**4**(1962), 25–39. MR**0152125**, https://doi.org/10.1137/1004004**[2]**Robert E. Barnhill,*An error analysis for numerical multiple integration. I*, Math. Comp.**22**(1968), 98–109. MR**0226852**, https://doi.org/10.1090/S0025-5718-1968-0226852-6**[3]**M. M. Chawla,*On the estimation of errors of Gaussian cubature formulas*, SIAM J. Numer. Anal.**5**(1968), 172–181. MR**0224280**, https://doi.org/10.1137/0705014**[4]**Philip Davis,*Errors of numerical approximation for analytic functions*, J. Rational Mech. Anal.**2**(1953), 303–313. MR**0054348****[5]**Philip J. Davis,*Interpolation and approximation*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR**0157156****[6]**I. A. Èzrohi, "General forms of remainder terms of linear formulae in multi-dimensional approximate analysis. I, II,"*Mat. Sb.*, v. 38(80), 1956, pp. 389-416; v. 43(85), 1957, pp. 9-28. (Russian) MR**18**, 32; MR**19**, 1199.**[7]**F. G. Lether,*Cross-Product Cubature Error Estimates*, Ph.D. Thesis, The University of Utah, Salt Lake City, Utah, 1969.**[8]**J. N. Lyness & B. J. J. McHugh, "Integration over multidimensional hypercubes. I: A progressive procedure,"*Comput. J.*, v. 6, 1963, pp. 264-270.**[9]**S. M. Nikol'skiĭ,*Quadrature Formulas*, Fizmatgiz, Moscow, 1958. (Russian)**[10]**Arthur Sard,*Linear approximation*, American Mathematical Society, Providence, R.I., 1963. MR**0158203****[11]**D. D. Stancu,*The remainder of certain linear approximation formulas in two variables*, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal.**1**(1964), 137–163. MR**0177240****[12]**Frank Stenger,*Error bounds for the evaluation of integrals by repeated Gauss-type formulae*, Numer. Math.**9**(1966), 200–213. MR**0205462**, https://doi.org/10.1007/BF02162084**[13]**A. H. Stroud and Don Secrest,*Gaussian quadrature formulas*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR**0202312**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0275673-6

Keywords:
Sard kernels,
Gaussian rules,
Peano kernels,
cross-product rules,
cubature error bounds

Article copyright:
© Copyright 1970
American Mathematical Society