aequationes mathematicae

, Volume 29, Issue 1, pp 183–203 | Cite as

Counterexamples in optimal quadrature

  • Arthur G. Werschulz
Research Papers

Abstract

It is widely believed that order of exactness is a good measure of the quality of an algorithm for numerical quadrature. We show that this is not the case, by exhibiting a situation in which the optimal algorithm does not even integrate constants exactly. We also show that there are situations in which the penalty for using equidistant nodes is unbounded. Finally, we show that the complexity of obtaining an ɛ-approximation can be an arbitrary function of ɛ, i.e., there is no hardest quadrature problem.

AMS (1980) subject classification

Primary 65D30 Secondary 68C25 

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References

  1. [1]
    Babenko, V. F.,Asymptotically sharp bounds for the remainder for the best quadrature formulas for several classes of functions. Math. Notes19 (1976), 187–193.Google Scholar
  2. [2]
    Bakhvalov, N. S.,Approximate computation multiple integrals (Russian). Vestmik Moscov. Univ. Ser. Mat. Meh. Astr. Fiz. Him. (1959), no. 4, 3–18.Google Scholar
  3. [3]
    Bakhvalov, N. S.,The optimality of linear operator approximation methods on convex function classes (Russian). Ž. Vyăisl. Mat. I Mat. Fiz.11 (1971), 1014–1018.Google Scholar
  4. [4]
    Lee, D. andTraub, J. F.,Optimal integration for functions of bounded variation. Research Report, Computer Science Department, Columbia University, New York, 1983.Google Scholar
  5. [5]
    Lorentz, G. G.,Approximation of functions. Holt, Rinehart and Winston, New York, 1966.Google Scholar
  6. [6]
    Motornyj, V. P., The best quadrature formula of the form\(\sum\limits_{k - 1}^n {p_k {\text{ }}f(x_k )}\) for some certain classes of periodic differential functions (Russian). Izv. Akad. Nauk SSSR Ser. Mat.38 (1974), 583–614.Google Scholar
  7. [7]
    Oden, J. T. andReddy, J. N.,An introduction to the mathematical theory of finite elements. Wiley-Interscience, New York, 1976.Google Scholar
  8. [8]
    Sard, A.,Linear approximation. American Mathematical Society, Providence, R.I., 1963.Google Scholar
  9. [9]
    Traub, J. F., Wasilkowski, G. andWoźniakowski, H.,Information, uncertainty, complexity. Addison-Wesley, Reading, Mass., 1983.Google Scholar
  10. [10]
    Traub, J. F. andWoźniakowski, H.,A general theory of optimal algorithms. Academic Press, New York-Toronto, 1980.Google Scholar

Copyright information

© Birkhäuser Verlag 1985

Authors and Affiliations

  • Arthur G. Werschulz
    • 1
    • 2
  1. 1.Division of Science and MathematicsFordham University/College at Lincoln CenterNew YorkUSA
  2. 2.Department of Computer ScienceColumbia UniversityNew YorkUSA

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