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Asymptotically optimal error bounds for quadrature rules of given degree

Author: H. Brass
Journal: Math. Comp. 61 (1993), 785-798
MSC: Primary 41A55; Secondary 65D32
MathSciNet review: 1192968
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Abstract: If the quadrature rule Q is applied to the function f, then the error can in many situations be bounded by $ {\rho _s}(Q){\left\Vert {{f^{(s)}}} \right\Vert _\infty }$, where $ {\rho _s}(Q)$ is independent of f. We obtain the asymptotics of these numbers for the Gaussian method $ Q_n^{\text{G}}\;(n = 1,2, \ldots )$ with very general weight functions and show that $ {\rho _s}(Q_n^{\text{G}})$ is (asymptotically) an upper bound for $ {\rho _s}(Q)$, if Q is any quadrature rule with the same degree as $ Q_n^{\text{G}}$.

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  • [1] H. Brass, Quadraturverfahren, Vandenhoeck und Ruprecht, Göttingen, 1977. MR 0443305 (56:1675)
  • [2] -, Restabschätzungen zur Polynomapproximation, Numerische Methoden der Approximationstheorie Bd. 7 (L. Collatz, G. Meinardus, and H. Werner, eds.), Birkhäuser Verlag, Basel, 1984.
  • [3] -, Eine Fehlerabschätzung für positive Quadraturformeln, Numer. Math. 47 (1985), 395-399. MR 808558 (86k:65019)
  • [4] -, Error bounds based on approximation theory, Numerical Integration--Recent Developments, Software and Applications (T. D. Espelid and A. Genz, eds.), NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 357, Kluwer, Dordrecht, 1992, pp. 147-163. MR 1198904 (94c:41036)
  • [5] H. Brass and K.-J. Förster, Error bounds for quadrature formulas near Gaussian quadrature, J. Comput. Appl. Math. 28 (1989), 145-154. MR 1038837 (91e:65037)
  • [6] H. Brass and G. Schmeisser, Error estimates for interpolatory quadrature formulae, Numer. Math. 37 (1981), 371-386. MR 627111 (82j:65012)
  • [7] G. Freud, Orthogonal polynomials, Pergamon Press, New York, 1971.
  • [8] G. Hämmerlin and K.-H. Hoffmann, Numerical mathematics, Springer-Verlag, New York, 1991. MR 1088482 (92d:65001)
  • [9] V. P. Motornyi, On the best quadrature formula of the form $ \sum\nolimits_{k = 1}^n {{p_k}f({x_k})} $ for some classes of differentiable periodic functions, Math. USSR Izv. 8 (1974), 591-620.
  • [10] K. Petras, Asymptotic behaviour of Peanokernels of fixed order, Numerical Integration III (H. Braßand G. Hämmerlin, eds.), Birkhäuser Verlag, Basel, 1988, pp. 186-198. MR 1021534 (91c:65019)
  • [11] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, 4th ed., Amer. Math. Soc., Providence, RI, 1975. MR 0310533 (46:9631)
  • [12] A. F. Timan, Theory of approximation of functions of a real variable, Macmillan, New York, 1963. MR 0192238 (33:465)

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Article copyright: © Copyright 1993 American Mathematical Society

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