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Zbl 1188.65068
Petković, Ljiljana D.; Petković, Miodrag S.; Džunić, Jovana
A class of three-point root-solvers of optimal order of convergence. (English)
[J] Appl. Math. Comput. 216, No. 2, 671-676 (2010). ISSN 0096-3003

Summary: The construction of a class of three-point methods for solving nonlinear equations of the eighth order is presented. These methods are developed by combining fourth order methods from the class of optimal two-point methods and a modified Newton's method in the third step, obtained by a suitable approximation of the first derivative based on interpolation by a nonlinear fraction. It is proved that the new three-step methods reach the eighth order of convergence using only four function evaluations, which supports the conjecture of {\it H. T. Kung} and {\it J. F. Traub} [J. Assoc. Comput. Mach. 21, 643--651 (1974; Zbl 0289.65023)] on the optimal order of convergence. Numerical examples for the selected special cases of two-step methods are given to demonstrate very fast convergence and a high computational efficiency of the proposed multipoint methods. Some computational aspects and the comparison with existing methods are also included.

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

Keywords: multipoint iterative methods; nonlinear equations; optimal order of convergence; computational efficiency; Kung-Traub's conjecture; Newton's method; numerical examples

Citations: Zbl 0289.65023

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