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Zbl 1161.65039
Bi, Weihong; Ren, Hongmin; Wu, Qingbiao
Three-step iterative methods with eighth-order convergence for solving nonlinear equations. (English)
[J] J. Comput. Appl. Math. 225, No. 1, 105-112 (2009). ISSN 0377-0427

For solving a nonlinear scalar equation, the authors propose a one parameter family of three-step iterative methods of eight-order. The new methods are based on the classical Newton method, on King's methods [see {\it R. F. King}, SIAM J. Numer. Anal. 10, 876--879 (1973; Zbl 0266.65040)], and on the methods of {\it C. Chun} and {\it Y. Ham} [Appl. Math. Comp. 193, 389--394 (2007)]. The iterations are expressed by means of divided difference of orders two and three and by means of a given real-valued function. To apply these methods for solving equations, three evaluations of the function from the left hand side of the equation and one evaluation of its first derivative are required. Using the definition of efficiency index, the family of derived methods has the index 1.682, better than the index of Newton's method, of King's methods and of Chun's methods. Numerical examples are performed and comparison of various iterative methods under the same total number of function evaluations are made.
[Iulian Coroian (Baia Mare)]

MSC 2000:: *65H05 Single nonlinear equations (numerical methods)
65Y20 Complexity and performance of numerical algorithms

Keywords: nonlinear equations; iterative methods; Newton's method; King's methods; order of convergence; index of efficiency; eighth-order convergence; numerical examples

Citations: Zbl 0266.65040

Cited in: Zbl 1190.65081

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