Memory in the iterative processes for nonlinear problems

Abstract

In this paper, we study different ways for introducing memory to a parametric family of optimal two-step iterative methods. We study the convergence and the stability, by means of real dynamics, of the methods obtained by introducing memory in order to compare them. We also perform several numerical experiments to see how the methods behave.

An Iterative Scheme for Evaluating Simultaneous Roots of Nonlinear Problems with Applications

“…(4) Considering as first step the Cordero et al scheme [13], which have four order of convergence and applying the proposed scheme, result an iterative scheme on eighth-order convergence, denoted by M 4t , as follows…”

“…The convergence of the proposed technique is proved in Section 3. We compare the results obtained with the proposed methods and with other known schemes like Newton's method, Ostrowki method [11], Jarratt's method [12], Cordero et al methods [13,14], Mir et al schemes [15,16], Shams et al method [17] and Petkovic et al scheme [2]. Several numerical experiments are performed in Section 4 by using the mentioned algorithms.…”

Iterative schemes for finding all roots simultaneously of nonlinear equations

“…In this section, we perform different numerical experiments in order to observe the behavior of the proposed methods. In this case, we modify, as discussed in the previous section, Newton's method, Steffensen's method [8], the N 4 and N 8 methods designed in [9], and the M 4 and M 6 schemes constructed in [10]. We denote these methods in the same way as in the previous section, that is, if the method is denoted by ϕ, then its variant with the added step is denoted by ϕ s .…”

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

“…Since analytical methods are not always suitable for finding solutions to equations, various numerical methods for solving nonlinear problems have been presented by scientists from different branches of science and engineering. Iterative methods are one of the best and most efficient methods to obtain solutions to nonlinear problems in mathematics [3][4][5][6][7][8].…”