# 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}
}```
• Published 1 October 1974
• Mathematics
• J. ACM
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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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
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This paper has the dual objectives of setting theoretical limits to the rates of convergence of iteration processes towards the zeros of a function when the values of the function and its derivatives are available and suggesting new families of computationally effective iteration formulas.
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Abstract : Let phi be an iteration for approximating the solution of a problem f. A new efficiency measure e(phi,f) is defined. For a given problem f, the authors define the optimal efficiency E(f)
A family of fourth order iterative methods for finding simple zeros of nonlinear functions is displayed. The methods require evaluation of the function and its derivative at the starting point of e...
In this paper we derive a simple and efficient fourth order multipoint iterative method for solving equations. Comparisons of computational efficiency are made with other well known techniques and a
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Algorithms based on Newton's interpolation formula are given for: simple polynomial interpolation, polynomial interpolation with derivatives supplied at some of the data points, interpolation with
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SIAM J. Comput.
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The relation between the goodness of an iteration algorithm and its new function evaluation and memory requirements are analyzed and a new conjecture is stated.