Optimal iterative processes for root-finding

Letf0(x) be a function of one variable with a simple zero atr0. An iteration scheme is said to be locally convergent if, for some initial approximationsx1, ...,xs nearr0 and all functionsf which are sufficiently close (in a certain sense) tof0, the scheme generates a sequence {xk} which lies nearr0 and converges to a zeror off. The order of convergence of the scheme is the infimum of the order of convergence of {xk} for all such functionsf. We study iteration schemes which are locally convergent and use only evaluations off,f′, ...,f[d] atx1, ...,xk−1 to determinexk, and we show that no such scheme has order greater thand+2. This bound is the best possible, for it is attained by certain schemes based on polynomial interpolation.