# A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve
## 3 Stability Analysis
### Fixed Points and Stability  
When the general class (20) is applied on polynomial \(p(z)\) with multiplicity \(m=2\), rational function \(M(z)\) is obtained depending of free complex parameter \(q_{\lambda}\), that we denote by \(\beta\). This rational operator is  
\[M(z)=\frac{N(z)}{(2x^{2}-3x-3)(3(\beta-3)+4(\beta-1)x^{4}-12(\beta-1)x^{3}+(3- 9\beta)x^{2}+2(7\beta-9)x)},\]  
where \(N(z)=-6\beta+8(\beta-1)x^{7}-4(7\beta-6)x^{6}-18(\beta-1)x^{5}+(65\beta-54)x^{4 }+(43\beta-59)x^{3}-9(3\beta-5)x^{2}+(77-37\beta)x+21\). In order to get bounded the set of converging seeds of the multiple root, a Mobius transformation is made, getting the conjugate rational function of \(M\) by \(h\), that we denote by \(MH\);  
\[MH(z,\beta)=\left(h\circ M\circ h^{-1}\right)(z),\]  
where \(h(z)=\frac{z+1}{z-1}\). It sends the multiple root to \(z=0\), the simple one to \(z=\infty\), and the divergence of the original scheme to \(z=1\). Operators \(M\) and \(MH\) have the same qualitative properties of stability. By solving equation \(MH(z,\beta)=z\), the fixed points of \(MH\) operator are obtained. Two of them, \(z=0\) and \(z=\infty\), come from the roots of polynomial \(p(z)\); the rest of them, if exist, are strange fixed points. The asymptotical behavior of all the fixed points (both multiple and simple, strange or not) plays a key role in the stability of the iterative methods involved, as the convergence to fixed points different from the roots means an important drawback for an iterative method; so, we proceed below with this analysis. To analyze the stability of the fixed points \(z_{F}\) of \(MH(z,\beta)\), we calculate its first derivative and evaluate it at every fixed point. Let us recall that the absolute value of this operator, \(|MH^{\prime}(z_{F},\beta)|\) is the stability function of fixed point \(z_{F}\). This stability function gives us information about the asymptotical behavior of \(z_{F}\). In general, the stability of other fixed points than the multiple root of \(p(z)\) depends on the value of parameter \(\beta\). From them, the following result can be stated. **Theorem 2**: _Rational function \(MH(z,\beta)\) has seven fixed points:_  
1. _Coming from the roots of_ \(p(z)\)_,_ \(z=0\) _is always superattracting and_ \(z=\infty\) _is a fixed point only if_ \(\beta\
eq-1\)_; moreover,_ \(z=\infty\) _is attracting only if_ \(\Re(\beta)>-\frac{1}{2}\)_. It is parabolic if_ \(\Re(\beta)=-\frac{1}{2}\) _and repulsive in other cases._
2. \(z=1\) _is a parabolic strange fixed point, for any_ \(\beta\in\mathbb{C}\)_._
3. _There exist also four strange fixed points, denoted by_ \(s_{i}(\beta)\)_,_ \(i=1,2,3,4\)_, corresponding to the roots of the fourth-degree polynomial_ \(q(t)=(2\beta+1)z^{4}+(34-28\beta)z^{3}+(57-30\beta)z^{2}+4(6\beta-13)z+8\)_. These points can be attracting in small sets of the area_ \([-1,4]\times[-2.5,2.5]\) _of the complex plane._  
Proof. : By solving equation \(MH(z,\beta)=z\), it is found that,  
\[-(z-1)^{2}z\Big{(}(2\beta+1)z^{4}+(34-28\beta)z^{3}+(57-30\beta)z^{2}+4(6\beta- 13)z+8\Big{)}=0.\]  
So, \(z=0\), \(z=1\) (double) and the roots of polynomial \(q(t)=(2\beta+1)z^{4}+(34-28\beta)z^{3}+(57-30\beta)z^{2}+4(6\beta-13)z+8\) are fixed points of \(MH(z,\beta)\). To check if \(z=\infty\) (coming from the simple root of \(p(z)\)) is a fixed point, the Inverse MH (IMH) operator  
\[IMH(z,\beta)=\frac{1}{MH(1/z,\beta)},\]  
is defined. Therefore, it can be checked that \(IMH(0,\beta)=0\) and then, \(z=\infty\) is a fixed point of \(MH(z,\beta)\). Now, the stability of these fixed points must be analyzed. It is straightforward to check that \(MH^{\prime}(0,\beta)=0\), so \(z=0\) is superattracting; moreover, \(MH^{\prime}(1,\beta)=1\) what yields to the parabolic character of \(z=1\) (the divergence of the original operator, previous to Mobius map).