# Convergence and Stability of a New Parametric Class of Iterative Processes for Nonlinear Systems
#### Multidimensional Real Dynamics Concepts  
Discrete dynamics is a very useful tool to study the stability of iterative schemes for solving nonlinear systems. An exhaustive description of this tool can be found in the book [8]. A resource used for the stability analysis of iterative schemes for solving nonlinear systems is to analyze the dynamical behavior of the vectorial rational operator obtained to apply the iterative expression on low degree polynomial systems. This technique generally uses quadratic or cubic polynomials [9].  
When we have scalar iterative processes, the tools to be used are of real or complex discrete dynamics. However, here, we handle a family of vectorial iterative methods, so real multidimensional dynamics must be used to analyze its stability, see [6]. We proceed by taking a system of quadratic polynomials on which we will apply our method in order to obtain the associated multidimensional rational operator and perform the analysis of the fixed and critical points in order to select members of the family with good stability.  
Some concepts used in this study are presented, see for instance [10].  
Let \(G:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) be the operator obtained from the iterative scheme on a polynomial system \(p(x)\). The set of successive images of \(x^{(0)}\) through \(G(x)\), \(\left\{x^{(0)},G\left(x^{(0)}\right),\ldots,G^{m}\left(x^{(0)}\right),\ldots\right\}\) is called the orbit of \(x^{(0)}\in\mathbb{R}^{n}\). \(x^{*}\in\mathbb{R}^{n}\) is a fixed point of \(G\) if \(G(x^{*})=x^{*}\). Of course, the roots of \(p(x)\) are a fixed point of \(G\), but there may be fixed points of \(G\) that are not solutions of system \(p(x)\). We refer them as strange fixed points. A point \(x\) that satisfies \(G^{k}(x)=x\) and \(G^{k-p}(x)\
eq x\), for \(p<k\) and \(k\geq 1\) is called a periodic point, of period \(k\). For classifying the stability of fixed or periodic points, we use the following result.  
**Theorem 1** ([8], pg. 558).: _Let \(G:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) be of type \(\mathcal{C}^{2}\). Assuming that \(x^{*}\) is a periodic \(k\)-point, \(k\geq 1\). If \(\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\) are the eigenvalues of \(G^{\prime}(x^{*})\), we have the following:_  
1. _If all eigenvalues_ \(\lambda_{k}\) _verify that_ \(|\lambda_{k}|<1\)_, then_ \(x^{*}\) _is an attracting point._
2. _If an eigenvalue_ \(\lambda_{k_{0}}\) _is such that_ \(\left|\lambda_{k_{0}}\right|>1\)_, then_ \(x^{*}\) _is unstable, that is, repulsor or saddle._
3. _If all eigenvalues_ \(\lambda_{k}\) _verify that_ \(|\lambda_{k}|>1\)_, then_ \(x^{*}\) _is a repulsive point._  
The set of preimages of any order of an attracting fixed point of the multidimensional rational function \(G\), \(x^{*}\),  
\[\mathcal{A}(x^{*})=\Big{\{}x^{(0)}\in\mathbb{R}^{n}:G^{m}\Big{(}x^{(0)}\Big{)} \to x^{*},m\rightarrow\infty\Big{\}},\]  
is the basin of attraction of \(x^{*}\), \(\mathcal{A}(x^{*})\).  
The solutions of \(G^{\prime}(x)=0\) are called the critical point of operator \(G\). The critical points different of the roots of \(p(x)\) are called a free critical point. The critical points are important for our analysis because of the following result from [PERSON] and [PERSON] (see [11; 12; 13]).  
**Theorem 2** ([PERSON] and [PERSON]).: _Let \(G\) be a rational function. The immediate basin of attraction of a periodic (or fixed) attractor point contains at least one critical point._