# Stability analysis of [PERSON] [PERSON]'s iterative method  
[PERSON]  
1 Department of Mathematics, Faculty of Science, Razi University, 67149 Kermanshah, Iran; [EMAIL_ADDRESS] (A.A.); [EMAIL_ADDRESS] (M.T.D.) [PERSON]  
2 Institute for Multidisciplinary Mathematics, Universitat Politecnica de Valencia, 46022 Valencia, Spain; [EMAIL_ADDRESS]  
[PERSON]  
1 Department of Mathematics, Faculty of Science, Razi University, 67149 Kermanshah, Iran; [EMAIL_ADDRESS] (A.A.); [EMAIL_ADDRESS] (M.T.D.) Footnote 1: email: {mdpi.com,mdpi. or the symmetric second-order one,  
\[[x^{(k)},y^{(k)};F]_{i,j}^{2}=\left(F_{i}(y_{1},y_{2},...,y_{j-1},y_{ j},x_{j+1},...,x_{n})-F_{i}(y_{1},y_{2},...,y_{j-1},x_{j},x_{j+1},...,x_{n})\right.\] \[\left.+F_{i}(x_{1},x_{2},...,x_{j-1},x_{j},y_{j+1},...,y_{n})-F_{i }(x_{1},x_{2},...,x_{j-1},y_{j},y_{j+1},...,y_{n})\right)/\left(2(y_{j}-x_{j}) \right), \tag{4}\]  
are proposed in [3]. Let us remark that operator (4) is symmetric and it can be used to evaluate the divided difference even when the problem is nonsymmetric. However, the number of evaluations of the scalar functions in the computation of (4) is higher than those in (3). Moreover, when divided difference (3) is used as an approximation for the Jacobian matrices appearing in any iterative method, then usually the iterative procedure does not preserve its order of convergence. The authors in [4] proposed to replace \(y=x+F(x)\) in (2) by \(y=x+\Gamma(x)\), being \(\Gamma(x)=(f_{1}(x)^{m},f_{2}(x)^{m},...,f_{n}(x)^{m})^{T}\), with \(m\in\mathbb{N}\). Then, divided difference  
\[[x,x+\Gamma(x);F]\Gamma(x)=F(x)-F(x+\Gamma(x)), \tag{5}\]  
becomes an approximation of \(F^{\prime}(x)\) of order \(m\). It was also shown that, by choosing a suitable value of \(m\), the order of convergence of any iterative method can be preserved with a reduced computational cost. So, the Jacobian-free variants of [PERSON]'s scheme that we analyze hold the second order of convergence of the original method. Our aim is to see if, further on the order of convergence of the method, the use of different divided differences to replace the Jacobian matrix in the iterative expression of [PERSON]'s method, can affect the dependence on initial estimations of the modified scheme to converge. By using [PERSON] expansion of the divided difference operator (5), authors in [4] proved the following results. **Theorem 1**. : _(See [4]) Let \(F\) be a nonlinear operator \(F:D\subseteq\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\) with coordinate functions \(f_{i}\), \(i=1,2,\ldots,n\) and \(m\in\mathbb{N}\) such that \(m\geq 1\). Let us consider the divided difference operator \([x+\Gamma(x),x;F]\), where \(\Gamma(x)=(f_{1}(x)^{m},f_{2}(x)^{m},...,f_{n}(x)^{m})^{T}\), then the order of the divided difference \([x+\Gamma(x),x;F]\) as an approximation of the Jacobian matrix \(F^{\prime}(x)\) is \(m\)._  
**Corollary 1**. : _(See [4]) Under the same assumptions as in Theorem 1, the order of the central divided difference operator_  
\[[x+\Gamma(x),x-\Gamma(x);F] \tag{6}\]  
_is \(2m\) as an approximation of \(F^{\prime}(x)\), being \(\Gamma(x)=(f_{1}(x)^{m},f_{2}(x)^{m},...,f_{n}(x)^{m})^{T}\)._  
Based on these results, in [4] it was presented a new technique to transform iterative schemes for solving nonlinear systems into Jacobian-free ones, preserving the order of convergence in all cases. The key fact of this new approach is the \(m\)th power of the coordinate functions of \(F(x)\), that needs different values depending on the order of convergence of the first step of the iterative method. This general procedure was checked, both theoretical and numerically, showing the preservation of the order of convergence and very precise results when the appropriate values of \(m\) were employed. Also the authors in [5] showed that the order of the approximation of \(F^{\prime}(x)\) might be improved (in terms of efficiency) by means of the Richardson extrapolation. It can be seen in the following result. **Lemma 1**.