Some iterative methods for solving a system of nonlinear equations

Stability Anomalies of Some Jacobian-Free Iterative Methods of High Order of Convergence

“…Taking these results into account, those fixed points whose components are ±1 and/or r 6 1 (β) for β < −118.1782 are attracting. In a similar way, all the fixed points whose components are ±1 and r 3 1 (β) for −0.5242 < β < −0.4991 are attracting. If a fixed point is composed exclusively by r 6 1 (β) and r 3 1 (β) out of these intervals of β, or by r 1 1 (β), r 2 1 (β), r 4 1 (β) and/or r 5 1 (β) for all values of β, it is repulsive.…”

“…In a similar way, all the fixed points whose components are ±1 and r 3 1 (β) for −0.5242 < β < −0.4991 are attracting. If a fixed point is composed exclusively by r 6 1 (β) and r 3 1 (β) out of these intervals of β, or by r 1 1 (β), r 2 1 (β), r 4 1 (β) and/or r 5 1 (β) for all values of β, it is repulsive. In any other case, the strange fixed point is classified as saddle.…”

“…There are no analytical techniques for solving these systems, so we approach their solutions by using iterative schemes. Although the most known iterative procedure is Newton's scheme, in recent years, the focus of this area of research has been in constructing new iterative methods, trying to improve Newton's one, in terms of convergence, efficiency, and stability (see, for example, some third-order schemes in References [1][2][3][4][5][6], or higher-order ones in References [7][8][9][10][11][12]). The key fact to get the most efficient methods is to evaluate as few Jacobian matrices as possible, per iteration (see Reference [13] and the references therein).…”

Local Convergence of a Family of Weighted-Newton Methods

“…Obviously, the function ϕ (x) is unbounded in the domain A. Therefore, the results in [5][6][7][8][9] and Method (2) cannot be applicable to such problems or its special cases that require the hypotheses on the third-or higher order derivatives of ϕ. Without a doubt, some of the iterative method in Brent [10] and Petkovíc et al [4] are derivative free and are used to locate zeros of functions.…”

A New Solution Approach for Non-Linear Equation Systems with Grey Wolf Optimizer

“…The solution methods are divided into two main categories direct methods and iterative methods. Several iterative methods have been developed to solve the nonlinear system of equations F(x) = 0, by using essentially Taylor's polynomial, decomposition, homotopy perturbation method, quadrature formulas and other techniques [3]. Iterative methods are divided into two groups called interval methods and continuation methods.…”