A New Three-Step Class of Iterative Methods for Solving Nonlinear Systems

Abstract

In this work, a new class of iterative methods for solving nonlinear equations is presented and also its extension for nonlinear systems of equations. This family is developed by using a scalar and matrix weight function procedure, respectively, getting sixth-order of convergence in both cases. Several numerical examples are given to illustrate the efficiency and performance of the proposed methods.

Development of a Family of Jarratt-Like Sixth-Order Iterative Methods for Solving Nonlinear Systems with Their Basins of Attraction

“…We require n scalar functions for each f and n 2 for each f . The concept of the efficiency index E applied to a nonlinear system of vector equations has been extended to treat the concept of computational efficiency by using CE = ρ 1/(d+op) [4], where op is the number of operations associated with products and quotients. Suppose that n is the size of the matrix needed in the nonlinear system of vector equations.…”

“…Since exact solutions for nonlinear equations are rarely available, we usually resort to their numerical solutions. To locate the desired numerical roots, many authors [1][2][3][4][5][6][7][8][9] have developed high-order iterative methods including optimal eighth-order ones [10][11][12][13][14][15].…”

“…The above nonlinear system is described in [4]. Selecting n = 10, we find d = 10 and solve (46) in R 10 with an initial guess vector x (0) = (0.75, 0.75, • • • , 0.75) T for the desired root x = (0.5149332, 0.5149332, • • • , 0.5149332) T ∈ R 10 in Table 8.…”

Convergence and Stability of a New Parametric Class of Iterative Processes for Nonlinear Systems

“…In Section 5, some numerical problems are considered to confirm the theoretical results. The proposed schemes for different values of parameter are considered and compared with Newton's method and some known sixth-order techniques, namely C6 1 , C6 2 , B6, PSH6 1 , PSH6 2 , XH6, introduced by Cordero et al in [3], Cordero et al in [4], Behl et al in [5], Capdevila et al in [6], and Xiao and Yin et al in [7].…”

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

“…In recent literature in this area of research, we can find other methods that reach higher orders (see, for example, previous studies [3][4][5][6][7] ), and they are designed by using different techniques like: Adomian decomposition, [8][9][10] multidimensional Steffensen-type schemes, [11][12][13][14][15][16] and weight function techniques. [17][18][19] Once a class of methods has been designed, it is interesting to carry out a multidimensional real dynamical study (see 20,21 ) in order to obtain the values most suitable of the parameters for setting stable schemes.…”