# Stability analysis of [PERSON] [PERSON]'s iterative method
: _(See [5]) Divided difference_  
\[R[x,x+\Gamma(x);F]:=\frac{1}{3}\left(2^{2}[x+\frac{1}{2}\Gamma(x),x-\frac{1}{2}\Gamma(x);F]-[x+\Gamma(x),x-\Gamma(x);F]\right), \tag{7}\]_which is obtained by Richardson extrapolation of (6) is an approximation of order \(4m\) of \(F^{\prime}(x)\)._  
Although the design and convergence analysis of iterative methods for solving nonlinear problems is a successful area of research in the last decades, it has been recently that the study of their stability has become usual (see, for example, [6; 7; 8; 9; 10]). So, when a method is presented, not only its order of convergence and efficiency are important, but also its dependance on the initial estimations used to converge. This is known as the stability analysis of the iterative scheme. The study of the stability of an iterative procedure has been mostly made by using techniques from complex discrete dynamics, that are very useful in the scalar case. Nevertheless, it frequently does not provide enough information when systems of nonlinear equations must be solved. This is the reason why the authors in [11] applied by first time real multidimensional discrete dynamics in order to analyze the performance of vectorial iterative methods on polynomial systems. In this way, it was possible to conclude about their stability properties: their dependence on the initial estimation used and the simplicity or complexity of the sets of convergent initial guesses (known as Fatou set) and their boundaries (Julia set). These procedure have been employed in the last years to analyze new and existing vectorial iterative schemes, see for instance [5; 12; 13; 14]. We are going to use these techniques in this paper to the vectorial rational functions obtained by applying different Jacobian-free variants of [PERSON]'s method on several low-degree polynomial systems. These vectorial rational functions are called also multidimensional fixed point functions in the literature. The polynomial systems used in this study are defined by the nonlinear functions:  
\[q(x)=\left\{ \begin{array}{l}q_{1}(x)=x_{1}^{2}-1\\ q_{2}(x)=x_{2}^{2}-1\end{array} \right.,\;\;r(x)=\left\{ \begin{array}{l}r_{1 }(x)=x_{1}^{6}-1\\ r_{2}(x)=x_{2}^{6}-1\end{array} \right.,\;\;p(x)=\left\{ \begin{array}{l}p_{1 }(x)=x_{1}^{2}-x_{2}-1\\ p_{2}(x)=x_{2}^{2}-x_{1}-1\end{array} \right..\]  
By using uncoupled systems as \(p(x)=0\) and \(r(x)=0\) and coupled as \(p(x)=0\), we can generalize the performance of the proposed methods to another nonlinear systems. Moreover, let us remark that these results can be obtained by using similar systems with size \(n>2\) but we analyze the case \(n=2\) in order to use two-dimensional plots to visualize the analytical findings. In the next section, the dynamical behavior of the fixed point functions of Jacobian-free versions of [PERSON]'s method applied on \(q(x)\), \(r(x)\) and \(p(x)\) are studied when forward, central and Richardson extrapolation-type of divided differences are used. To get this aim, some dynamical concepts must be introduced. Definition 1: (See [11]) Let \(G:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\) be a vectorial function. The orbit of the point \(x^{(0)}\in\mathbb{R}^{n}\) is defined as the set of successive images of \(x^{(0)}\) by the vectorial function, \(\{x^{(0)},G(x^{(0)}),...,G^{m}(x^{(0)}),...\}\). The dynamical behavior of the orbit of a point of \(\mathbb{R}^{n}\) can be classified depending on its asymptotic behavior. In this way, a point \(x^{*}\in\mathbb{R}^{n}\) is a fixed point of \(G\) if \(G(x^{*})=x^{*}\). The following results are well known results in discrete dynamics and in this paper we use them to study the stability of nonlinear operators. Theorem 2: (See [15]) Let \(G:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) be \(\mathcal{C}^{2}\). Assume that \(x^{*}\) is a \(k\)-periodic point, \(k\geq 1\). Let \(\lambda_{1},\lambda_{2},\cdots,\lambda_{n}\) be the eigenvalues of \(G^{\prime}(x^{*})\). 1. If all eigenvalues \(\lambda_{j}\) have \(|\lambda_{j}|<1\), then \(x^{*}\) is attracting. 2.