A FAMILY OF TWO-POINT METHODS WITH MEMORY FOR SOLVING NONLINEAR EQUATIONS

An efficient family of two-point derivative free methods withmemory for solving nonlinear equations is presented. It is provedthat the convergence order of the proposed family is increasedfrom 4 to at least $2+sqrt{6}approx 4.45,$ 5,$fraccc12igl(5+sqrt{33},igr)approx 5.37$ and 6, depending onthe accelerating technique. The increase of convergence order isattained using a suitable accelerating technique by varying a freeparameter in each iteration. The improvement of convergence rateis achieved without any additional function evaluations meaningthat the proposed methods with memory are very efficient.Moreover, the presented methods are more efficient than allexisting methods known in literature in the class of two-pointmethods and three-point methods of optimal order eight. Numericalexamples and the comparison with the existing two-point methodsare included to confirm theoretical results and highcomputational efficiency.