Abstract
An efficient family of two-point derivative free methods with memory for solving nonlinear equations is presented. It is proved that the convergence order of the proposed family is increased from 4 to at least 2 + ?6 ? 4.45, 5, 1/2 (5 + ?33) ? 5.37 and 6, depending on the accelerating technique. The increase of convergence order is attained using a suitable accelerating technique by varying a free parameter in each iteration. The improvement of convergence rate is achieved without any additional function evaluations meaning that the proposed methods with memory are very efficient. Moreover, the presented methods are more efficient than all existing methods known in literature in the class of two-point methods and three-point methods of optimal order eight. Numerical examples and the comparison with the existing two-point methods are included to confirm theoretical results and high computational efficiency. 2010 Mathematics Subject Classification. 65H05
Highlights
The main goal and motivation in constructing iterative methods for solving nonlinear equations is to attain as high as possible order of convergence with minimal computational cost
The highest possible computational efficiency of these methods is closely connected to the hypothesis of Kung and Traub [6] from 1974
They have conjectured that the order of convergence of any multipoint method without memory, consuming n function evaluations per iteration, cannot exceed the bound 2n−1
Summary
An efficient family of two-point derivative free methods with memory for solving nonlinear equations is presented. It is proved that the√convergence o1rd5e+r o√f 3th3e proposed family is increased ≈ 5.37 and 6, depending on from 4 to at least 2 + 6 ≈ the accelerating technique. 4.45, 5, The increase of convergence order is attained using a suitable accelerating technique by varying a free parameter in each iteration. The improvement of convergence rate is achieved without any additional function evaluations meaning that the proposed methods with memory are very efficient. The presented methods are more efficient than all existing methods known in literature in the class of two-point methods and three-point methods of optimal order eight. Numerical examples and the comparison with the existing two-point methods are included to confirm theoretical results and high computational efficiency
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