# Stability analysis of [PERSON] [PERSON]'s iterative method
## 2 Jacobian-Free Variants of [PERSON]'s Method
### Second-Degree Polynomial System \(p(x)=0\)  
The results about \(\mathrm{FMN}_{m}\), \(\mathrm{CMN}_{m}\) and \(\mathrm{RMN}_{m}\) classes applied on second-degree polynomial system \(p(x)=0\) are stated in the following propositions. They resume, in each case, the iteration function, the fixed and critical points. **Proposition 7**. : _The coordinate functions of the fixed point operator \(k^{1,m}(x)\) associated to \(F\mathrm{MN}_{m}\) on \(p(x)\), for \(m=1,2,\ldots\), are_  
\[k_{1}^{1,m}(x)=x_{1}-\frac{-1-x_{1}+x_{2}^{2}}{-1+\frac{(-x_{1}^{ 2}+(x_{1}+(1-x_{1}^{2}+x_{2})^{m}))(-x_{2}^{2}+(x_{2}+(1+x_{1}-x_{2}^{2})^{m}) )^{2}}{(1-x_{1}^{2}+x_{2})^{m}(1+x_{1}-x_{2}^{2})^{m}}}\] \[-\frac{(-1+x_{1}^{2}-x_{2})(-x_{2}^{2}+(x_{2}+(1+x_{1}-x_{2}^{2}) ^{m})^{2})}{(1+x_{1}-x_{2}^{2})^{m}(-1+\frac{(-x_{1}^{2}+(x_{1}+(1-x_{1}^{2}+x _{2})^{m})^{2})(-x_{2}^{2}+(x_{2}+(1+x_{1}-x_{2}^{2})^{m})^{2})}{(1-x_{1}^{2}+x _{2})^{m}(1+x_{1}-x_{2}^{2})^{m}}})},\;\;m=1,2,\cdots\] \[k_{2}^{1,m}(x)=x_{2}-\frac{-1+x_{1}^{2}-x_{2}}{-1+\frac{(-x_{1}^ {2}+(x_{1}+(1-x_{1}^{2}+x_{2})^{m}))(-x_{2}^{2}+(x_{2}+(1+x_{1}-x_{2}^{2})^{m}) ^{2})}{(1-x_{1}^{2}+x_{2})^{m}(1+x_{1}-x_{2}^{2})^{m}}}\] \[-\frac{(-1-x_{1}+x_{2}^{2})(-x_{1}^{2}+(x_{1}+(1-x_{1}^{2}+x_{2}) ^{m})^{2})}{(1-x_{1}^{2}+x_{2})^{m}(-1+\frac{(-x_{1}^{2}+(x_{1}+(1-x_{1}^{2}+ x_{2})^{m})^{2})(-x_{2}^{2}+(x_{2}+(1+x_{1}-x_{2}^{2})^{m})^{2})}{(1-x_{1}^{2}+x_{2} )^{m}(1+x_{1}-x_{2}^{2})^{m}}})},\;\;m=1,2,\cdots.\]  
_Moreover,_  
1. _For_ \(m\geq 1\) _the only fixed points are the roots_ \((-1,0)\)_,_ \((0,-1)\)_,_ \(\left(\frac{1}{2}\left(1-\sqrt{5}\right),\frac{1}{2}\left(1-\sqrt{5}\right)\right)\) _and_ \(\left(\frac{1}{2}\left(1+\sqrt{5}\right),\frac{1}{2}\left(1+\sqrt{5}\right)\right)\)_, that are superattracting._
2. _For_ \(m=1\)_, the free critical points are_ \((-0.632026,-0.460233)\)_,_ \((-0.460233,-0.632026)\)_,_ \((-0.203287,0.294659)\) _and_ \((0.294659,-0.203287)\)_._  
The calculation of free critical points of \(k^{1,m}(x)\) for \(m>1\) is very complicated hence, so it has been provided only for \(m=1\). Figure 8 shows the dynamical planes of the fixed point function \(k^{1,m}(x)\) for \(m=1,2,3,4\). Similar results to polynomial system \(q(x)=0\) and \(r(x)=0\) have been obtained in this case, as by increasing \(m\), the wideness of the basins of attraction decreases. To study the behavior of the vectorial function in the black area of dynamical planes, we visualize the diverging orbits of \(k^{1,m}(x)\) with the starting point \((0,-3)\), after 200 iterations. The vectors on the top of the Figure 8a-d, show the last vectorial iterate with the starting point \((0,-3)\). Figure 7: Dynamical plane for \(\mathrm{RMN}_{m}\) method, \(m=1,2,3,4\) on \(r(x)\). Green, orange, red and blue areas are the basins of attraction of the roots of \(q(x)\) (marked as white stars); black area denotes divergence and white squares are free critical points. The following proposition shows, that using \(\mathrm{CMN}_{m}\), for \(m=1,2,\ldots\), and \(\mathrm{RMN}_{m}\), for \(m=1,2,\ldots\), on polynomial system \(p(x)=0\), we obtain, as was the case for the system \(q(x)=0\), the same fixed point function as the classical [PERSON]'s method. **Proposition 8**. : _The coordinates of the fixed point operator \(k^{2,m}(x)\) (which is denoted by \(k^{2}(x)\)) associated to \(\mathrm{CMN}_{m}\) for \(m=1,2,\ldots\) and \(\mathrm{RMN}_{m}\) for \(m=1,2,\ldots\) on polynomial system \(p(x)\) are_  
\[k_{1}^{2}(x) = \frac{1+2(1+x_{1}^{2})x_{2}+x_{2}^{2}}{-1+4x_{1}4x_{2}},\] \[k_{2}^{2}(x) = \frac{1+x_{1}^{2}+2x_{1}(1+x_{2}^{2})}{-1+4x_{1}4x_{2}},\]  
_which are the same components of the fixed point function of [PERSON]'s method on \(p(x)\). Moreover, the fixed points of \(k^{2}(x)\) are the roots \((-1,0)\), \((0,-1)\), \(\left(\frac{1}{2}\left(1-\sqrt{5}\right),\frac{1}{2}\left(1-\sqrt{5}\right)\right)\) and \(\left(\frac{1}{2}\left(1+\sqrt{5}\right),\frac{1}{2}\left(1+\sqrt{5}\right)\right)\) that are superattracting.