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.
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References
Brent, R.P.: Algorithms for Minimization without Derivatives. Englewood Cliffs, New Jersey: Prentice Hall 1973
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, 1976
Sikorski, K., Trojan, J.: Asymptotic Optimality of the Bisection Method (in progress)
Schwartz, L.: Analyse Mathematique. Paris: Herman 1967
Traub, J.F., Wozniakowski, H.: A General Theory of Optimal Algorithms. New York: Academic Press 1980
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Sikorski, K. Bisection is optimal. Numer. Math. 40, 111–117 (1982). https://doi.org/10.1007/BF01459080
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DOI: https://doi.org/10.1007/BF01459080