# Optimal Order of One-Point and Multipoint Iteration

@article{Kung1974OptimalOO, title={Optimal Order of One-Point and Multipoint Iteration}, author={H. T. Kung and Joseph F. Traub}, journal={J. ACM}, year={1974}, volume={21}, pages={643-651} }

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<supscrpt><italic>n</italic>-1</supscrpt> which use <italic>n</italic> evaluations of ƒ and no derivative evaluations, as well as a second family of iterations of order 2<supscrpt><italic>n</italic>-1</supscrpt> based on <italic>n</italic> — 1 evaluations of ƒ and one of ƒ′. In particular, with four evaluations an iteration of eighth order is constructed. The…

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