An optimal fourth order method for solving nonlinear equations

Abstract

In this paper, we use both weight functions and composition techniques together for solving non-linear equations. We designed a new fourth order iterative method to increase the order of convergence without increasing the functional evaluations in a drastic way. This method uses one evaluation of the function and two evaluations of the first derivative. The new method attains the optimality with efficiency index 1.587. The convergence analysis of our new methods is discussed. Furthermore, the correlations between the attracting domains and the corresponding required number of iterations have also been illustrated and discussed. The comparison with several numerical methods and the use of complex dynamics and basins of attraction show that the new method gives good results.

An Optimal Class of Fourth-order Iterative Methods without Restraint on the First Derivative

“…Furthermore Hafiz et al also developed the following optimal fourth-order method known as (HKM) [9], where a=2, b=4 and, c=-5:…”

“…While traditional approaches are unable to solve problems of this kind, a diversity of iterative approaches have been developed to reach a simple zero of the function f (x) = 0 where, f : I ⊆ R → R for an open interval I. A well-known approach for solving nonlinear equations and the most widely used to f (x) = 0 is Newton's method (NM), see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and their references. The conventional Newton's scheme can be written as:…”

“…According to the Kung-Traub conjecture [21] which is given by 2 τ −1 , wherein an iterative process (NM) is an optimal method. Various modifications have been made to Newton's scheme to increase its convergence rate [2][3][4][5][6][7][8][9][10][11][12][13][17][18][19][20][21][22][23][24][25]. However, the majority of these methods overlooked the fact that during the iterative process, the first derivative disappears.…”

Convergence and Stability of Optimal Two-step Fourth-order and its Expanding to Sixth Order for Solving Nonlinear Equations

“…In numerical analysis, many methods produce sequences of real numbers such as the iterative methods for solving nonlinear equations. At times, the sequences' convergence is slow and their use in solving practcal problems is limited a fast convergent one [9].…”

An optimal fourth-order second derivative free iterative method for nonlinear scientific equations

“…Acording to this conjecture, an iterative method is said to be an optimal one if it needs (n + 1) evaluations per iteration and posses convergence order 2 n . Some useful optimal fourth-order iterative methods have been constructed by various researchers (Sharma et al, 2020;Ali et al, 2020;Shams et al, 2020;Cordero et al, 2021;Hafiz and Khirallah, 2021). Cordero et al, (2010) introduced the following optimal fourth-order method:…”