derwo-stepwith-memory adaptivemethodbaseon Hansen-Patrick An eighth-order two-step with-memory adaptive method base on Hansen-Patrick'sdynamicmethodand its dynamic

A tri-parametric family of two-point iterative methods with six-order convergence for solving tions of the given function per iteration It is optimal in the sense of the Kung and Traub conj nonlinearequations has been proposed.Each derivative-free methodmemberofthe family requires only on these methods with-memory methods with convergence orders of 6 7 7 53 and 8 are const arameters ofthe self-accelerator are calculatedusing Newton interpolation andinformationfro three evaluations of the given function per iteration. It is optimal in the sense of the Kung and Traub tandprevious terations The proposedfamiy has an efficiency index of196 an2Num conjecture. Based on these methods, with-memory methods with convergence orders of 6, 7, 7.53, and arsons have beenmade to reveal the highefficiency ofthe developedmethodThe dynamicalst 8 are constructed. The parameters of the self-accelerator are calculated using Newton interpolation and ve schemes reflects a good overview of their stability information from the current and previous iterations. The proposed family has an efficiency index of 1.96 convergence roperties and graphical aspe and 2. Numerical comparisons have been made to reveal g tt a ti b ns in th mpl pla Al h y dynamical study of iterative schemes reflects a good overview of their stability, convergence properties, and t the be t i ht f ti th t ha th l t tt tio ba i fo diffe t l n mial g u graphical aspects by drawing attraction d Ad ti th d ith y w r p ve m o m m behavior of new methods to select the best o s polynomials. , the high efficiency o v xamin d the d f the developed method. The i b h i f n basins g e p o y in the complex plane. Also, we have exami A l t i f t ned the dynamic i N li y, cc er or p a me e , B s n a weight function that has the largest attractio a o n n n basins for different