Optimal iterative processes for root-finding

Let f0(x) be a function of one variable with a simple zero at r0. An iteration scheme is said to be locally convergent if, for some initial approximations x1, ..., xs near r0 and all functions f which are sufficiently close (in a certain sense) to f0, the scheme generates a sequence {xk} which lies near r0 and converges to a zero r of f. The order of convergence of the scheme is the infimum of the order of convergence of {xk} for all such functions f. We study iteration schemes which are locally convergent and use only evaluations of f, f′, ..., f[d] at x1, ..., xk-1 to determine xk, and we show that no such scheme has order greater than d+2. This bound is the best possible, for it is attained by certain schemes based on polynomial interpolation. © 1972 Springer-Verlag.