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A family of two-point methods with memory for solving nonlinear equations

Petković Miodrag S. (Faculty of Electronic Engineering, Department of Mathematics, Niš)
Džunić Jovana (Faculty of Electronic Engineering, Department of Mathematics, Niš)
Petković Ljiljana D. (Faculty of Mechanical Engineering, Niš)

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

Keywords: Nonlinear equations, Iterative methods, Multipoint methods with memory, Acceleration of convergence, Computational efficiency