# Stability analysis of [PERSON] [PERSON]'s iterative method
## 2 Jacobian-Free Variants of [PERSON]'s Method
### Second-Degree Polynomial System \(q(x)=0\)  
**Proposition 2**. : _The coordinate functions of the fixed point operator \(\lambda^{1,m}(x)\) associated to \(\text{FMN}_{m}\) for \(m=1,2,\ldots\) on polynomial system \(q(x)=0\) are_  
\[\lambda_{j}^{1,m}(x)=x_{j}-\frac{q_{j}(x)}{2x_{j}+q_{j}(x)^{m}},\ \ j=1,2.\]  
_Moreover,_  
1. _For_ \(m=1,2,...\)_, the only fixed points are the roots of_ \(q(x)\)_,_ \((-1,-1)\)_,_ \((-1,1)\)_,_ \((1,-1)\) _and_ \((1,1)\)_, that are also superattracting. There is no strange fixed point in this case._
2. _The components of free critical points_ \(c_{\lambda^{1,m}}^{n}=(k,l)\) _are roots of_ \(2q_{j}(x)+(2+2m)x_{j}q_{j}(x)^{m}+q_{j}(x)^{(2m)}=0\)_, provided that_ \(k\) _and_ \(l\) _are not equal to_ \(1\) _and_ \(-1\)_, simultaneously._  
**Remark 1**. : _Except for \(m=1\), that is a case with 12 free critical points, for \(m>1\) there exist 32 free critical points of the fixed point operator associated to \(\text{FMN}_{m}\). In particular, free critical points for the fixed point function \(\lambda^{1,m}(x)\), \(m=1,2,\ldots,6\) are_  
\[(k,l),\ k,l\in\{-3.73205,-0.267949,-1,1\},\ \ \text{for}\ \ m=1,\] \[(k,l),\ k,l\in\{-2.11724,-1.13867,0.191305,0.760289,1,-1\},\ \ \text{for}\ \ m=2,\] \[(k,l),\ k,l\in\{-1.85429,-1.2072,-0.614069,-0.14308,1,-1\},\ \ \text{for}\ \ m=3,\] \[(k,l),\ k,l\in\{-1.74245,-1.24306,0.112834,0.541487,-1,1\},\ \ \text{for}\ \ m=4,\] \[(k,l),\ k,l\in\{-1.67929,-1.26615,-0.496325,-0.0927087,1,-1\},\ \ \text{for}\ \ m=5,\] \[(k,l),\ k,l\in\{-1.63822,-1.28266,0.0785163,0.464271,1,-1\},\ \ \text{for}\ \ m=6,\]  
_provided that \(k\) and \(l\) are not equal to 1 and \(-1\), simultaneously._  
Figure 1 shows the dynamical behavior of fixed point function \(\lambda^{1,m}(x)\) for \(m=1,2,\ldots,6\). These figures have been obtained by the routines described in [16]. To draw them, a mesh of \(400\times 400\) points has been used, 200 was the maximum number of iterations involved and \(10^{-3}\) the tolerance used as stopping criterium. In this paper, we have used a white star to show the roots of the nonlinear polynomial system and a white square for the free critical points. Figure 1 shows that, as greater \(m\) is, the wideness of basins of attraction decreases, in spite of having a better approximation of the Jacobian matrix. The color assigned to each of the basin of attraction corresponds to one of the roots of the nonlinear system. The black area shows the no convergence in the maximum number of iterations, or divergence. To see the behavior of the vectorial function in the black area of dynamical planes, we visualize the orbit of the rational function corresponding to the starting point \((-3,-3)\) after 200 iterations. This orbit appears as a yellow circle per each iterate and yellow lines between each pair of consecutive iterates. In Figure 1a, corresponding to \(m=1\), its value in the 200 th iteration is \((-213.0141,-213.0141)\). Figure 1b, which corresponds to \(m=2\), shows lower rate of divergence (or convergence to infinity), being its value at the 200 th iteration \((-8.6818,-8.6818)\). This effect is higher by increasing \(m\) but, for \(m\geq 3\) it is observed that  
\[\frac{\|\lambda^{1,m}(x^{(0)})-\lambda^{1,m}(x^{(200)})\|}{\|\lambda^{1,m+1}( x^{(0)})-\lambda^{1,m+1}(x^{(200)})\|}\approx 1.\]  
The vectors on the top of Figure 1c-f, corresponds to the last iterate with the starting point \(x^{(0)}=(-3,-3)\). **Proposition 3**. : _The coordinates of the fixed point operator \(\lambda^{2,m}(x)\) associated to \(\text{CMN}_{m}\), for \(m=1,2,\ldots\), and \(\text{RMN}_{m}\), for \(m=1,2,\ldots\), on polynomial system \(q(x)=0\) are_  
\[\lambda_{j}^{2}(x)=\frac{1+x_{j}^{2}}{2x_{j}},\ \ j=1,2,\]  
_which are the same components of the fixed point function of [PERSON]'s method on \(q(x)\)._  
Figure 1: Dynamical plane for \(\text{FMN}_{m}\), \(m=1,2,\ldots,6\) on \(q(x)\).