### Optimal Order of One-Point and Multipoint IterationOptimal Order of One-Point and Multipoint Iteration

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 Author Kung, H. T. ♦ Traub, J. F. Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©1974 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Abstract The problem is to calculate a simple zero of a nonlinear function ƒ by iteration. There is exhibited a family of iterations of order $2\textit{n}-1$ which use $\textit{n}$ evaluations of ƒ and no derivative evaluations, as well as a second family of iterations of order $2\textit{n}-1$ based on $\textit{n}$ — 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 $\textit{optimal}$ order of one general class of multipoint iterations is $2\textit{n}-1$ and that an upper bound on the order of a multipoint iteration based on $\textit{n}$ evaluations of ƒ (no derivatives) is $2\textit{n}.It$ is conjectured that a multipoint iteration without memory based on $\textit{n}$ evaluations has optimal order $2\textit{n}-1.$ ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 1974-10-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 21 Issue Number 4 Page Count 9 Starting Page 643 Ending Page 651

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Source: ACM Digital Library