University of Tabriz Computational Methods for Differential Equations 2345-3982 10 4 2022 10 01 A two-step method adaptive with memory with eighth-order for solving nonlinear equations and its dynamic 1007 1026 14153 10.22034/cmde.2022.46651.1961 EN Vali Torkashvand Department of Mathematics, Shahre-Qods Branch, Islamic Azad University, Tehran, Iran. Department of Basic Science, Shahid Sattari Aeronatuical Uinversity of Science and Technology, Tehran, Iran. Journal Article 2021 06 22 In this work, we have constructed the with memory two-step method with four convergence degrees by entering the maximum self-accelerator parameter(three parameters). Then, using Newton’s interpolation, a with-memory method with a convergence order of 7.53 is constructed. Using the information of all the steps, we will improve the convergence order by one hundred percent, and we will introduce our method with convergence order 8. Numerical examples demonstrate the exceptional convergence speed of the proposed method and confirm theoretical results. Finally, we have presented the dynamics of the adaptive method and other without-memory methods for complex polynomials of degrees two, three, and four. The basins of attraction of existing with-memory methods are present and compared to illustrate their performance. https://cmde.tabrizu.ac.ir/article_14153_d220454953782bfebf8ee9950f03bae3.pdf