Numerische Mathematik

, Volume 40, Issue 1, pp 111–117

Bisection is optimal

  • K. Sikorski
Article

Summary

We seek an approximation to a zero of a continuous functionf:[a,b]→ℝ such thatf(a)≦0 andf(b)≧0. It is known that the bisection algorithm makes optimal use ofn function evaluations, i.e., yields the minimal error which is (b−a)/2n+1, see e.g. Kung [2]. Traub and Wozniakowski [5] proposed using more general information onf by permitting the adaptive evaluations ofn arbitrary linear functionals. They conjectured [5, p. 170] that the bisection algorithm remains optimal even if these general evaluations are permitted. This paper affirmatively proves this conjecture. In fact we prove optimality of the bisection algorithm even assuming thatf is infinitely many times differentiable on [a, b] and has exactly one simple zero.

Subject Classfications

AMS(MOS): 65H10 CR: 5.15 

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References

  1. 1.
    Brent, R.P.: Algorithms for Minimization without Derivatives. Englewood Cliffs, New Jersey: Prentice Hall 1973Google Scholar
  2. 2.
    Kung, H.T.: The Complexity of Obtaining Starting Points for Solving Operator Equations by Newton's Method. In: Analitic Computational Complexity. Traub, J.F. (ed.), New York: Academic Press pp. 35–57, 1976Google Scholar
  3. 3.
    Sikorski, K., Trojan, J.: Asymptotic Optimality of the Bisection Method (in progress)Google Scholar
  4. 4.
    Schwartz, L.: Analyse Mathematique. Paris: Herman 1967Google Scholar
  5. 5.
    Traub, J.F., Wozniakowski, H.: A General Theory of Optimal Algorithms. New York: Academic Press 1980Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • K. Sikorski
    • 1
  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  2. 2.University of WarsawPoland

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