Bisection is not optimal on the average

SummaryWe seek the zero of a continuous increasing functionf: [0, 1] → [−1, 1] such thatf(0)=−1 andf(1)=1. It is known that the bisection method makes optimal use ofn function evaluations within a worst case analysis. In this paper we study the average error with respect to the natural measure of Graf et al. (1986). We prove that the bisection method is not optimal on the average. Actually, the average error of the bisection method is about (1/2)n, while the average error of the optimal method is less than αn with some α<1/2.