Optimal Order of One-Point and Multipoint Iteration

The problem is to calculate a simple zero of a nonlinear function ƒ by iteration. There is exhibited a family of iterations of order 2 n -1 which use n evaluations of ƒ and no derivative evaluations, as well as a second family of iterations of order 2 n -1 based on n — 1 evaluations of ƒ and one of ƒ′. In particular, with four evaluations an iteration of eighth order is constructed. The best previous result for four evaluations was fifth order. It is proved that the optimal order of one general class of multipoint iterations is 2 n -1 and that an upper bound on the order of a multipoint iteration based on n evaluations of ƒ (no derivatives) is 2 n . It is conjectured that a multipoint iteration without memory based on n evaluations has optimal order 2 n -1 .