# Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations
## 2 Convergence of the New Family
}\frac{f^{(q)}(\xi)}{f^{\prime}(\xi)}\), \(q=2,3,...\)  
Dividing (3) by (4), we get  
\[\frac{f(x_{k})}{f^{\prime}(x_{k})}=e_{k}-C_{2}e_{k}^{2}+2\Big{(}C_{2}^{2}-C_{ 3}\Big{)}e_{k}^{3}-\Big{(}4C_{2}^{3}-7C_{2}C_{3}+3C_{4}\Big{)}e_{k}^{4}+ \mathcal{O}\Big{(}e_{k}^{5}\Big{)}. \tag{5}\]  
Replacing (5) in the first step of family (1), we have  
\[y_{k}=\xi+C_{2}e_{k}^{2}-2\Big{(}C_{2}^{2}-C_{3}\Big{)}e_{k}^{3}+\Big{(}4C_{2} ^{3}-7C_{2}C_{3}+3C_{4}\Big{)}e_{k}^{4}+\mathcal{O}\Big{(}e_{k}^{5}\Big{)}. \tag{6}\]  
Using Taylor expansion again, similar to (3), to develop \(f(y_{k})\) around \(\xi\), we get  
\[f(y_{k})=f^{\prime}(\xi)\Big{[}C_{2}e_{k}^{2}-2\Big{(}C_{2}^{2}-C_{3}\Big{)}e _{k}^{3}+\Big{(}5C_{2}^{3}-7C_{2}C_{3}+3C_{4}\Big{)}e_{k}^{4}+\mathcal{O} \Big{(}e_{k}^{5}\Big{)}\Big{]}. \tag{7}\]  
With (3), (6), and (7), we calculate the divided difference operator defined in (2), obtaining  
\[f[x_{k},y_{k}]=f^{\prime}(\xi)\Big{[}1+C_{2}e_{k}+\Big{(}C_{2}^{2}+C_{3}\Big{)} e_{k}^{2}-\Big{(}2C_{2}^{3}-3C_{2}C_{3}-C_{4}\Big{)}e_{k}^{3}+\mathcal{O} \Big{(}e_{k}^{4}\Big{)}\Big{]}. \tag{8}\]  
Then, substituting (3), (4), (6), and (8) in the second step of family (1), we have  
\[z_{k}=\xi+\Big{(}C_{2}^{3}-C_{2}C_{3}\Big{)}e_{k}^{4}+\mathcal{O}\Big{(}e_{k} ^{5}\Big{)}. \tag{9}\]  
Using Taylor series once again, similar to (3), to expand \(f(z_{k})\) around \(\xi\), we get  
\[f(z_{k})=f^{\prime}(\xi)\Big{[}\Big{(}C_{2}^{3}-C_{2}C_{3}\Big{)}e_{k}^{4}+ \mathcal{O}\Big{(}e_{k}^{5}\Big{)}\Big{]}. \tag{10}\]  
Replacing (4) and (8) in \(u_{k}\) and \(v_{k}\) of family (1), we have  
\[=f^{\prime}(\xi)+f^{\prime\prime}(\xi)e_{k}+\frac{1}{2!}f^{\prime \prime\prime}(\xi)e_{k}^{2}+\frac{1}{3!}f^{(iv)}(\xi)e_{k}^{3}+\mathcal{O}(e_{k }^{4})\]0  
\[=f^{\prime}(\xi)+f^{\prime\prime}(\xi)e_{k}+\frac{1}{2!}f^{\prime \prime\prime}(\xi)e_{k}^{2}+\frac{1}{3!}f^{(iv)}(\xi)e_{k}^{3}+\mathcal{O}(e_{k }^{4})\]1  
Finally, substituting (4), (9)-(12) in the third step of family (1), we get  
\[=f^{\prime}(\xi)+f^{\prime\prime}(\xi)e_{k}+\frac{1}{2!}f^{\prime \prime\prime}(\xi)e_{k}^{2}+\frac{1}{3!}f^{(iv)}(\xi)e_{k}^{3}+\mathcal{O}(e_{k }^{4})\]2  
being the error equation  
\[=f^{\prime}(\xi)+f^{\prime\prime}(\xi)e_{k}+\frac{1}{2!}f^{\prime \prime\prime}(\xi)e_{k}^{2}+\frac{1}{3!}f^{(iv)}(\xi)e_{k}^{3}+\mathcal{O}(e_{k }^{4})\]3  
and the proof is finished. From Theorem 1, it follows that the new triparametric family of iterative methods has an order of convergence of four for any real or complex values of the parameters \(\alpha\), \(\beta\), and \(\gamma\). However, convergence can be speed-up if only one parameter is held and the family is reduced to an uniparametric iterative scheme. The latter can be seen in Theorem 2. Theorem 2: _Let \(f:I\subseteq\mathbb{R}\rightarrow\mathbb{R}\) be a sufficiently differentiable function on an open interval \(I\) and \(\xi\in I\) a simple root of the nonlinear equation \(f(x)=0\). Suppose that \(f(x)\) is continuous and sufficiently differentiable in an environment of the simple root \(\xi\), and \(x_{0}\) is an initial estimate close enough to \(\xi\). Then, the sequence \(\{x_{k}\}_{k\geq 0}\) obtained by using the expression (1) converges to \(\xi\) with an order of convergence of six, provided that \(\beta=1+\alpha\) and \(\gamma=1-\alpha\), being its error equation_  
\[=f^{\prime}(\xi)+f^{\prime\prime}(\xi)e_{k}+\frac{1}{2!}f^{\prime \prime\prime}(\xi)e_{k}^{2}+\frac{1}{3!}f^{(iv)}(\xi)e_{k}^{3}+\mathcal{O}(e_{k }^{4})\]4  
_where \(e_{k}=x_{k}-\xi\), \( C_{q}=\dfrac{1}{q! }\dfrac{f^{(q)}(\xi)}{f^{\prime}(\xi)}\) and \(q=2,3,...\)_  
Proof. : Let \(\xi\) be a simple root of \(f(x)\) (that is, \(f(\xi)=0\) and \(f^{\prime}(\xi)\
eq 0\)) and \(x_{k}=\xi+e_{k}\). Using Taylor expansion of \(f(x_{k})\) and \(f^{\prime}(x_{k})\) around \(\xi\), we have  
\[