# Convergence and Stability of a New Parametric Class of Iterative Processes for Nonlinear Systems
## 3 Stability Analysis  
Let us consider a polynomial system of \(n\) variables \(q(x)=0\), \(q:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) where \(q_{i}(x)=x_{i}^{2}-1\), \(i=1,2,...,n\) and we denote by \(K(x)\) the associated rational function. From now on, we denote by \(ACTV^{6}(x,\alpha)=\big{(}actv_{1}^{6}(x,\alpha),actv_{2}^{6}(x,\alpha),\dots, actv_{n}^{6}(x,\alpha)\big{)}\) the vectorial function obtained when class (9) is applied on \(q(x)\). As \(q(x)=0\) is uncoupled, all functions \(actv_{j}^{6}(x,\alpha)\) are analogous, with the difference of the index \(j=1,2,...,n\). Their expressions are  
\[actv_{j}^{6}(x,\alpha)=\frac{p(x,\alpha)}{128x_{j}^{5}\Big{(}1+x_{j}^{2}\Big{)} ^{2}\Big{(}1+3x_{j}^{2}\Big{)}},\quad j=1,2,...,n, \tag{19}\]  
where,  
\[p(x,\alpha) = 1+x_{j}^{6}(404-20\alpha)+x_{j}^{10}(782-6\alpha)+\alpha+x_{j}^{ 12}(77+\alpha)-2x_{j}^{2}(1+3\alpha)\] \[+5x_{j}^{8}(155+3\alpha)+x_{j}^{4}(11+15\alpha).\]  
There are values of \(\alpha\) for which the operator coordinates are simplified; we show the particular case when \(\alpha=1\). \[actv_{j}^{6}(x,1)=\frac{1-7x_{j}^{2}+34x_{j}^{4}+90x_{j}^{6}+125x_{j}^{8}+13x_ {j}^{10}}{64x_{j}^{5}\Big{(}1+x_{j}^{2}\Big{)}^{2}},\quad j=1,2,...,n. \tag{20}\]  
By determining and analyzing the corresponding fixed points of the operator, we present a synthesis of the most relevant results. **Theorem 4**. : _Roots of \(q(x)\) are the components of \(2^{n}\) superattracting fixed points of \(ACTV^{6}(x,\alpha)\) associated to the class of iterative methods (9). The same happens with the roots of \(l(t)=-1-\alpha+(1+5\alpha)t^{2}-(10+10\alpha)t^{4}+(10\alpha-286)t^{6}-(421+5 \alpha)t^{8}+(\alpha-307)t^{10}\) depending on \(\alpha\): (a) If \(\alpha<-1\) or \(\alpha>307\), there are two real roots of \(l(t)\), denoted by \(l_{i}(\alpha)\), \(i=1,2\). Fixed points \((l_{\sigma_{1}}(\alpha),l_{\sigma_{2}}(\alpha),\dots,l_{\sigma_{n}}(\alpha))\) where \(\sigma_{i}\in\{1,2\}\), are repulsive points. However, if any of \(l_{\sigma_{i}}(\alpha)=\pm 1\), \(j\in\{1,2,\dots,n\}\), then they are saddle points. (b) \(ACTV^{6}(x,\alpha)\) has no strange fixed points for \(-1\leq\alpha\leq 307\)._  
Figure 2: Efficiency index IO for systems of size \(n=10\) to \(n=50\). Proof. : To calculate the fixed points of \(ACTV^{6}(x,\alpha)\), we solve \(actv^{6}_{j}(x,\alpha)=x_{j}\),  
\[-\Big{(}x_{j}^{2}-1\Big{)}l(t)=-1-\alpha+(1+5\alpha)t^{2}-(10+10 \alpha)t^{4}+(10\alpha-286)t^{6}-(421+5\alpha)t^{8} \tag{21}\] \[+(\alpha-307)t^{10},\]  
for \(j=1,2,\ldots,n\), that is, \(x_{j}=\pm 1\) and roots of \(l(t)\), provided that \(t\
eq 0\). At most, two of the roots of \(l(t)\) are not complex, depending on \(\alpha\). The qualitative performance of \(ACTV^{6}(x,\alpha)\) is deduced from the eigenvalues of \(ACTV^{6}(x,\alpha)\) evaluated at the fixed points. Due to the nature of the polynomial system, these eigenvalues coincide with the coordinate function of the rational operator:  
\[ \begin{split}& Eig_{j}(l_{j}(\alpha),\ldots,l_{j}(\alpha))=\\ &\frac{\big{(}-1+l_{j}(\alpha)^{2}\big{)}^{5}\big{(}5(1+\alpha)+ 3l_{j}(\alpha)^{6}(77+\alpha)+3l_{j}(\alpha)^{4}(65+17\alpha)+l_{j}(\alpha)^{ 2}(49+37\alpha)\big{)}}{128l_{j}(\alpha)^{6}\big{(}1+l_{j}(\alpha)^{2}\big{)} ^{3}\big{(}1+3l_{j}(\alpha)^{2}\big{)}^{2}}\end{split}  \tag{22}\]  
We calculate the absolute values of these eigenvalues only where fixed points are real; it is clear that those fixed points \(l_{j}(\alpha)=\pm 1\) are super attracting. We proceed to plot some of the eigenvalues; if \(\alpha<-1\), the eigenvalues of \(\big{(}ACTV^{6}\big{)}^{\prime}(x,\alpha)\) at any strange fixed point are named saddle points when their combinations have some component +1,-1 and the combinations of real roots coming from \(l(t)\) are named repulsors because all eigenvalues are greater than 1 (see Figure 3a); if \(\alpha>307\), a similar behavior is observed (see Figure 3b).