# Stability analysis of [PERSON] [PERSON]'s iterative method
## 2 Jacobian-Free Variants of [PERSON]'s Method
### Sixth-Degree Polynomial System \(r(x)=0\)  
The corresponding results about \(\mathrm{FMN}_{m}\), \(\mathrm{CMN}_{m}\) and \(\mathrm{RMN}_{m}\) classes applied on sixth-degree polynomial system \(r(x)=0\) are summarized in the following propositions. They resume, in each case, the iteration function, the fixed and critical points. **Proposition 4**. : _The coordinate functions of the fixed point operator \(h^{1,m}(x)\) associated to \(\mathrm{FMN}_{m}\) on \(r(x)\), for \(m=1,2,\ldots\), are_  
\[h^{1,m}_{j}(x)=x_{j}-\frac{r_{j}(x)^{1+m}}{-x_{j}^{6}+(x_{j}+r_{j}(x)^{m})^{6} },\;\;j=1,2.\]  
_Moreover,_  
1. _For_ \(m\geq 1\) _the fixed points are_ \((-1,-1)\)_,_ \((-1,1)\)_,_ \((1,-1)\) _and_ \((1,1)\) _that are also superattracting. There are no strange fixed points in this case._
2. _The coordinates of free critical points_ \(c^{n}_{h^{1,m}}=(k,l)\) _are the roots of the polynomial_ \[(x_{j}^{6}-(x_{j}+r_{j}(x)^{m})^{6})^{2} + r_{j}(x)^{1+m}(-6x_{j}^{5}+6(1+6 mx_{j}^{5}r_{j}(x)^{-1+m})(x_{j} +r_{j}(x)^{m})^{5})\] \[+ 6(1+m)x_{j}^{5}r_{j}(x)^{m}(x_{j}^{6}-(x_{j}+r_{j}(x)^{m})^{6}),\] _for_ \(j=1,2\) _and_ \(m=1,2,\ldots\)_, provided that_ \(k\) _and_ \(l\) _are not equal to_ \(-1\) _and_ \(1\) _simultaneously._  
Figure 2: Dynamical plane of \(\mathrm{CMN}_{m}\), \(\mathrm{RMN}_{m}\) and [PERSON]’s method on \(q(x)\). Green, orange, red and blue areas are the basins of attraction of the roots of \(q(x)\) (marked as white stars). **Remark 2**. : _Except for \(m=1\), in which case that the fixed point function \(h^{1,m}(x)\) has \(12\) free critical points, the multidimensional rational function \(\text{FMN}_{m}\), for \(m=2,3,\ldots\), has \(32\) free critical points. The free critical points in this case are_  
\[(k,l),\;\;k,l\in\{-1.3191,-0.290716,-1,1\}\;\;\text{for}\;\;m=1,\] \[(k,l),\;\;k,l\in\{-1.20592,-1.01655,0.289432,0.978576,-1,1\}\;\; \text{for}\;\;m=2,\] \[(k,l),\;\;k,l\in\{-1.17593,-1.03983,-0.934077,-0.288186,-1,1\}\;\; \text{for}\;\;m=3,\] \[(k,l),\;\;k,l\in\{-1.16205,-1.0543,0.286976,0.897504,-1,1\}\;\; \text{for}\;\;m=4,\] \[(k,l),\;\;k,l\in\{-1.15401,-1.064,-0.868592,-0.285801,-1,1\}\;\; \text{for}\;\;m=5,\] \[(k,l),\;\;k,l\in\{-1.14876,-1.07101,0.284659,0.845166,-1,1\}\;\; \text{for}\;\;m=6,\]  
_provided that \(k\) and \(l\) are not equal to \(-1\) and \(1\) simultaneously._  
Figure 3 shows the dynamical planes of the fixed point function \(h^{1,m}(x)\) for \(m=1,2,\ldots,6\). Inside the black areas, there are regions where the orbits tend toward the poles of the rational function (those points that make the denominator of the elements of the rational function null). In fact, the orbits of points in these areas reach very close to the roots, so the vectorial rational function at these points become numerically singular. Figure 3a,b show two points in these black areas. \[(x_{j}^{6}-(x_{j}+r_{j}(x)^{m})^{6})^{2} + r_{j}(x)^{1+m}(-6x_{j}^{5}+6(1+6 mx_{j}^{5}r_{j}(x)^{-1+m})(x_{j} +r_{j}(x)^{m})^{5})\]0 _Cont._Figures 4 and 5 show details of the dynamical planes of \(\mathrm{FMN}_{2}\) and \(\mathrm{FMN}_{3}\) methods. These figures show the difference between the basins of attraction of an odd member with an even member of the family \(\mathrm{FMN}_{m}\). Because of the symmetry, the transpose of the basin of attraction of \((1,-1)\) coincide with that of \((-1,1)\), so we only depicted one of them. \[(x_{j}^{6}-(x_{j}+r_{j}(x)^{m})^{6})^{2} + r_{j}(x)^{1+m}(-6x_{j}^{5}+6(1+6 mx_{j}^{5}r_{j}(x)^{-1+m})(x_{j} +r_{j}(x)^{m})^{5})\]1 Basins of attraction of \((-1,-1)\), \((-1,1)\) and \((1,1)\) for \(\mathrm{FMN}_{2}\) method 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. \[(x_{j}^{6}-(x_{j}+r_{j}(x)^{m})^{6})^{2} + r_{j}(x)^{1+m}(-6x_{j}^{5}+6(1+6 mx_{j}^{5}r_{j}(x)^{-1+m})(x_{j} +r_{j}(x)^{m})^{5})\]2 Dynamical plane of \(\mathrm{FMN}_{m}\) method for \(m=1,2,\ldots,6\) on \(r(x)\).