Optimal iterative processes for root-finding

Abstract

Letf ₀(x) be a function of one variable with a simple zero atr ₀. An iteration scheme is said to be locally convergent if, for some initial approximationsx ₁, ...,x _s nearr ₀ and all functionsf which are sufficiently close (in a certain sense) tof ₀, the scheme generates a sequence {x _k} which lies nearr ₀ and converges to a zeror off. The order of convergence of the scheme is the infimum of the order of convergence of {x _k} for all such functionsf. We study iteration schemes which are locally convergent and use only evaluations off,f′, ...,f ^[d] atx ₁, ...,x _k−1 to determinex _k, 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.