A class of three-point root-solvers of optimal order of convergence |
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Authors: | Ljiljana D Petkovi? Miodrag S Petkovi? |
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Institution: | a Faculty of Mechanical Engineering, Department of Mathematics, University of Niš, 18000 Niš, Serbia b Faculty of Electronic Engineering, Department of Mathematics, University of Niš, 18000 Niš, Serbia |
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Abstract: | 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 Kung-Traub conjecture 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. |
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Keywords: | Multipoint iterative methods Nonlinear equations Optimal order of convergence Computational efficiency Kung-Traub&rsquo s conjecture |
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