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Derivative free two-point methods with and without memory for solving nonlinear equations. (English) Zbl 1200.65034
Summary: Two families of derivative free two-point iterative methods for solving nonlinear equations are constructed. These methods use a suitable parametric function and an arbitrary real parameter. It is proved that the first family has the convergence order four requiring only three function evaluations per iteration. In this way it is demonstrated that the proposed family without memory supports the Kung-Traub hypothesis [{\it H. T. Kung} and {\it J. F. Traub}, J. Assoc. Comput. Mach. 21, 643--651 (1974; Zbl 0289.65023)] on the upper bound $2^n$ of the order of multipoint methods based on $n + 1$ function evaluations. Further acceleration of the convergence rate is attained by varying a free parameter from step to step using information available from the previous step. This approach leads to a family of two-step self-accelerating methods with memory whose order of convergence is at least $2 + \sqrt{5} \approx 4.236$ and even $2 + \sqrt{6} \approx 4.449$ in special cases. The increase of convergence order is attained without any additional calculations so that the family of methods with memory possesses a very high computational efficiency. Numerical examples are included to demonstrate exceptional convergence speed of the proposed methods using only few function evaluations.

##### MSC:
 65H05 Single nonlinear equations (numerical methods)
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##### References:
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