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Zbl 1166.65338
Ren, Hongmin; Wu, Qingbiao; Bi, Weihong
A class of two-step Steffensen type methods with fourth-order convergence. (English)
[J] Appl. Math. Comput. 209, No. 2, 206-210 (2009). ISSN 0096-3003

Summary: Based on Steffensen's method, we derive a one-parameter class of fourth-order methods for solving nonlinear equations. In the proposed methods, an interpolating polynomial is used to get a better approximation to the derivative of the given function. Each member of the class requires three evaluations of the given function per iteration. Therefore, this class of methods has efficiency index which equals 1.587. {\it H. T. Kung} and {\it J. F. Traub} conjectured an iteration using $n$ evaluations of $f$ or its derivatives without memory is of convergence order at most $2n-1$ [J. Assoc. Comput. Mach. 21, 643--651 (1974; Zbl 0289.65023)]. The new class of fourth-order methods agrees with the conjecture of Kung-Traub for the case $n=3$. Numerical comparisons are made to show the performance of the presented methods.

MSC 2000:: *65H05 Single nonlinear equations (numerical methods)

Keywords: nonlinear equations; iterative methods; Steffensen's method; derivative free method; order of convergence; the conjecture of Kung-Traub

Citations: Zbl 0289.65023

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Zentralblatt MATH Berlin [Germany]