New fourth- and sixth-order classes of iterative methods for solving systems of nonlinear equations and their stability analysis

Abstract

In this paper, a two-step class of fourth-order iterative methods for solving systems of nonlinear equations is presented. We further extend the two-step class to establish a new sixth-order family which requires only one additional functional evaluation. The convergence analysis of the proposed classes is provided under several mild conditions. A complete dynamical analysis is made, by using real multidimensional discrete dynamics, in order to select the most stable elements of both families of fourth and sixth-order of convergence. To get this aim, a novel tool based on the existence of critical points has been used, the parameter line. The analytical discussion of the work is upheld by performing numerical experiments on some application oriented problems. Finally on the basis of numerical results, it has been concluded that our methods are comparable with the existing ones of similar nature in terms of order and efficiency and also that the stability results provide the most efficient member of each class of iterative schemes.Keywords Systems of nonlinear equations • Order of convergence • Multipoint iterative methods • Stability analysis.

A new sixth-order Jarratt-type iterative method for systems of nonlinear equations

“…Now, we want to verify the numerical results of our iterative method. For this purpose, we consider some examples and compare the results of our scheme, namely, F S4, F S5, and F S6, with respect to number of iterations n, absolute residual error of the corresponding function in F X (n) , and absolute error in two consecutive iterations X (n) −X (n−1) that are given in Tables 3, 4 Example 3.2 We take a 3 × 3 system F 1 (X ) of nonlinear equations from [6], such that…”

“…The error equation is given as e n+1 = 1 729 (−54c 2 2 + 9c 3 + 8Q (1)c 2 2 )(32P (1)c 3 2 + 3c 3 2 + 81c 2 c 3 − 9c 4 )e 6 n +O(e 7 n ).…”

An Efficient Jarratt-Type Iterative Method for Solving Nonlinear Global Positioning System Problems

“…In addition, n 2 products are computed in the case where a matrix is multiplied by a vector. We compared our schemes ( 11)-( 13) for EI and CEI (Figure 7) with the sixth-order methods given in [10,12,13]. These schemes are given below.…”

“…However, not all methods are extendable in this way. Recently, researchers have proposed sixth-order iterative methods using weight functions and parameters [10][11][12][13]. We typically turn to their numerical solutions because exact nonlinear solutions are rarely available.…”

Isonormal surfaces: A new tool for the multidimensional dynamical analysis of iterative methods for solving nonlinear systems

“…Due to the existence of critical points in the immediate basin of attraction of any attracting fixed or periodic point (see 26 ), the analysis of the asymptotic behavior of free critical points gives us relevant information about the performance of the rational operator and the related class of iterative methods. To study the orbits of critical points, we use the parameter line, firstly introduced in.…”