# A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve
## 3 Stability Analysis  
In the previous section, we have designed a modification of Chebyshev-Halley scheme able to find multiple roots of nonlinear equations, holding the third-order of convergence of their original partners. Our aim in this section is to check the role of the parameter of the new class of iterative method in the stability of the resulting scheme: are these methods capable to find also single roots? how far can be the initial estimation in order to assure convergence? are these situations different depending on the value of \(q_{\sigma}\)? that is, is the stability different for the members of the set of parametric procedures?  
In order to arrange this analysis, we apply our proposed schemes on the nonlinear function \(p(z)=(z-1)^{m}(z+1)\), with \(z=1\) as multiple root of multiplicity \(m\) and a simple root at \(z=-1\). In all cases, \(\alpha=1\), although qualitatively similar results are found for other values of \(\alpha\). This is the most simple nonlinear function containing two roots, one simple and one \(m\)-multiple. Although the results cannot be directly extrapolated to any nonlinear function, several analysis on different nonlinear problems confirm, in the numerical section, these results.  
First, we introduce some dynamical terms mentioned in this paper (see, for instance, [17]). Let \(R:\widehat{\mathbb{C}}\longrightarrow\widehat{\mathbb{C}}\) be a rational function, where \(\widehat{\mathbb{C}}\) is the Riemann sphere, the orbit of a point \(z_{0}\) is given as:  
\[\{z_{0},R(z_{0}),R^{2}(z_{0}),\ldots,R^{n}(z_{0}),\ldots\},\]  
where the \(k\)-th composition of the map \(R\) with itself is denoted by \(R^{k}\).  
We analyze map \(R\) by classifying the starting points from the asymptotical performance of the orbits. In these terms, a point \(z_{0}\) is called fixed point of \(R\) if \(R(z_{0})=z_{0}\); it is a periodic point of period \(p>1\) if \(R^{p}(z_{0})=z_{0}\) and \(R^{k}(z_{0})\
eq z_{0}\), for \(k<p\).  
On the other hand, a point \(z_{0}\) is called critical point of \(R\) if \(R^{\prime}(z_{0})=0\). The asymptotic behavior of the critical points is a key fact for analyzing the stability of the method: a classical result from [PERSON] and [PERSON] (see, for instance, [18]) each immediate basin of attraction holds at least one critical point, that is, in the connected component of the basin of attraction holding the attractor there is also a critical point.  
Moreover, a fixed point of \(R\), \(z_{0}\), is said to be attracting if \(|R^{\prime}(z_{0})|<1\), or superattracting if \(|R^{\prime}(z_{0})|=0\); it is repulsive if \(|R^{\prime}(z_{0})|>1\) and parabolic if \(|R^{\prime}(z_{0})|=1\).  
Indeed, when \(R\) depends also on one or several parameters, then \(|R^{\prime}(z_{0},\alpha_{1},\ldots,\alpha_{k})|\) is not a scalar, but a function of \(\alpha_{i}\), for \(i=1,2,\ldots,k\). Then, \(|R^{\prime}(z_{0},\alpha_{1},\ldots,\alpha_{k})|\) is called stability function of the fixed point and it gives us the character of the fixed point in terms of the value of \(\alpha_{i}\), for \(i=1,2,\ldots,k\).  
On the other hand, let us also remark that when fixed and critical points are found such that they are not equivalent to the roots of the polynomial \(p(z)\), then they are called strange fixed and free critical points, respectively.  
The basin of attraction of an attractor \(\alpha\) is defined as:  
\[\mathcal{A}(\alpha)=\{z_{0}\in\widehat{\mathbb{C}}:R^{n}(z_{0})\to\alpha,n \to\infty\}.\]  
The Fatou set of the rational function \(R\), is the set of points \(z\in\widehat{\mathbb{C}}\) whose orbits tend to an attractor (fixed point or periodic point). Its complement in \(\widehat{\mathbb{C}}\) is the Julia set, \(\mathcal{J}(R)\). So the basin of attraction of any fixed point belongs to the Fatou set and the boundary of the basin of attraction belongs to the Julia set.