# Bisection is optimal

- Received:

DOI: 10.1007/BF01459080

- Cite this article as:
- Sikorski, K. Numer. Math. (1982) 40: 111. doi:10.1007/BF01459080

## Summary

We seek an approximation to a zero of a continuous function*f*:[*a,b*]→ℝ such that*f(a)*≦0 and*f(b)*≧0. It is known that the bisection algorithm makes optimal use of*n* function evaluations, i.e., yields the minimal error which is (*b−a*)/2^{n+1}, see e.g. Kung [2]. Traub and Wozniakowski [5] proposed using more general information on*f* by permitting the adaptive evaluations of*n 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 that*f* is infinitely many times differentiable on [*a, b*] and has exactly one simple zero.