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Zbl 1195.65067
Sharma, Janak Raj; Sharma, Rajni
A new family of modified Ostrowski's methods with accelerated eighth order convergence. (English)
[J] Numer. Algorithms 54, No. 4, 445-458 (2010). ISSN 1017-1398; ISSN 1572-9265/e

Summary: Based on Ostrowski's fourth order method, we derive a family of eighth order methods for the solution of nonlinear equations. In terms of computational cost, the family requires three function evaluations and one evaluation of the first derivative. Therefore, the efficiency index of the present methods is 1.682 which is better than the efficiency index 1.587 of Ostrowski's method. {\it H. T. Kung} and {\it J. F. Traub} [J. Assoc. Comput. Mach. 21, 643--651 (1974; Zbl 0289.65023)] conjectured that multipoint iteration methods without memory based on $n$ evaluations have optimal order $2^{n - 1}$. Thus, the family agrees with the Kung-Traub conjecture for the case $n = 4$. The efficacy of the present methods is tested on a number of numerical examples. It is observed that our methods are competitive with other similar robust methods and very effective in high precision computations.

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

Keywords: nonlinear equations; Newton's method; convergence acceleration; efficiency; numerical examples

Citations: Zbl 0289.65023

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