Bisection is optimal
- Cite this article as:
- Sikorski, K. Numer. Math. (1982) 40: 111. doi:10.1007/BF01459080
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 . Traub and Wozniakowski  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.