# Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations
## 2 Convergence of the New Family
\begin{split} f(z_{k})&=f^{\prime}(\xi)\Big{[} \Big{(}C_{2}^{3}-C_{2}C_{3}\Big{)}e_{k}^{4}-2\Big{(}2C_{2}^{4}-4C_{2}^{2}C_{3}+ C_{3}^{2}+C_{2}C_{4}\Big{)}e_{k}^{5}+\\ &\quad\Big{(}10C_{2}^{5}-30C_{2}^{3}C_{3}+12C_{2}^{2}C_{4}-7C_{3 }C_{4}+3C_{2}\Big{(}6C_{3}^{2}-C_{5}\Big{)}\Big{)}e_{k}^{6}+\mathcal{O}\Big{(} e_{k}^{7}\Big{)}\Big{]}.\end{split}  \tag{22}\]  
Replacing (16) and (20) in \(u_{k}\) and \(v_{k}\) of family (1), we have  
\[ \begin{split} u_{k}&=C_{2}e_{k}-\Big{(}3C_{2}^{2}-2 C_{3}\Big{)}e_{k}^{2}+\Big{(}8C_{2}^{3}-10C_{2}C_{3}+3C_{4}\Big{)}e_{k}^{3}+ \Big{(}-20C_{2}^{4}+37C_{2}^{2}C_{3}-\\ &\quad 8C_{2}^{3}-14C_{2}C_{4}+4C_{5}\Big{)}e_{k}^{4}+\Big{(}48C_{2}^ {5}-118C_{2}^{3}C_{3}+51C_{2}^{2}C_{4}-22C_{3}C_{4}+\\ &\quad C_{2}\Big{(}55C_{3}^{2}-18C_{5}\Big{)}+5C_{6}\Big{)}e_{k}^ {5}+\mathcal{O}\Big{(}e_{k}^{6}\Big{)},\end{split}  \tag{23}\]  
\[ \begin{split} v_{k}&=1+C_{2}e_{k}-2\Big{(}C_{2}^{2}-C_ {3}\Big{)}e_{k}^{2}+3\Big{(}C_{2}^{3}-2C_{2}C_{3}+C_{4}\Big{)}e_{k}^{3}+\Big{(} -3C_{2}^{4}+11C_{2}^{2}C_{3}-\\ &\quad 4C_{3}^{2}-8C_{2}C_{4}+4C_{5}\Big{)}e_{k}^{4}+\Big{(}-10C_{2}^ {3}C_{3}+14C_{2}^{2}C_{4}+C_{2}\Big{(}11C_{3}^{2}-10C_{5}\Big{)}+\\ &\quad 5(-2C_{3}C_{4}+C_{6})\Big{)}e_{k}^{5}+\mathcal{O}\Big{(}e_{ k}^{6}\Big{)}.\end{split}  \tag{24}\]  
Finally, substituting (16) and (21)-(24) in the third step of family (1), we get  
\[ \begin{split} x_{k+1}&=\xi+(1-\alpha-\gamma)C_{2} \Big{(}C_{2}^{2}-C_{3}\Big{)}e_{k}^{4}+\Big{(}(-4+6\alpha-\beta+5\gamma)C_{2}^ {4}+\\ &\quad(8-10\alpha+\beta-9\gamma)C_{2}^{2}C_{3}-2(1-\alpha-\gamma)C _{3}^{2}-2(1-\alpha-\gamma)C_{2}C_{4}\Big{)}e_{k}^{5}+\\ &\quad\Big{(}(10-22\alpha+9\beta-14\gamma)C_{2}^{5}-(30-53\alpha+ 15\beta-39\gamma)C_{2}^{3}C_{3}+\\ &\quad 2(6-8\alpha+\beta-7\gamma)C_{2}^{2}C_{4}-7(1-\alpha-\gamma)C _{3}C_{4}+\\ &\quad C_{2}\Big{(}(18-25\alpha+4\beta-21\gamma)C_{3}^{2}-3(1- \alpha-\gamma)C_{5}\Big{)}\Big{)}e_{k}^{6}+\mathcal{O}\Big{(}e_{k}^{7}\Big{)}, \end{split}  \tag{25}\]being the error equation  
\[e_{k+1} =(1-\alpha-\gamma)C_{2}\Big{(}C_{2}^{2}-C_{3}\Big{)}e_{k}^{4}+\Big{(} (-4+6\alpha-\beta+5\gamma)C_{2}^{4}+\] \[\quad(8-10\alpha+\beta-9\gamma)C_{2}^{2}C_{3}-2(1-\alpha-\gamma)C _{3}^{2}-2(1-\alpha-\gamma)C_{2}C_{4}\Big{)}\epsilon_{k}^{5}+\] \[\quad\Big{(}(10-22\alpha+9\beta-14\gamma)C_{2}^{5}-(30-53\alpha+1 5\beta-39\gamma)C_{2}^{3}C_{3}+ \tag{26}\] \[\quad 2(6-8\alpha+\beta-7\gamma)C_{2}^{2}C_{4}-7(1-\alpha-\gamma)C _{3}C_{4}+\] \[\quad C_{2}\Big{(}(18-25\alpha+4\beta-21\gamma)C_{3}^{2}-3(1- \alpha-\gamma)C_{5}\Big{)}\Big{)}e_{k}^{6}+\mathcal{O}\Big{(}\epsilon_{k}^{7} \Big{)}.\]  
To cancel the factors accompanying \(e_{k}^{4}\) and \(e_{k}^{5}\) in (26), it must be satisfied that \(\alpha+\gamma=1\), \(6\alpha-\beta+5\gamma=4\) and \(10\alpha-\beta+9\gamma=8\). It is easy to show that this system of equations has infinite solutions for  
\[\beta=1+\alpha\quad\text{and}\quad\gamma=1-\alpha, \tag{27}\]  
where \(\alpha\) is a free parameter. Therefore, replacing (27) in (26), we obtain  
\[e_{k+1}=\Big{(}6C_{2}^{5}-7C_{2}^{3}C_{3}+C_{2}C_{3}^{2}\Big{)}e_{k}^{6}+ \mathcal{O}\Big{(}\epsilon_{k}^{7}\Big{)}, \tag{28}\]  
and the proof is finished. From Theorem 2, it follows that, if we only hold parameter \(\alpha\) in (1), the new triparametric family of iterative methods is reduced to an uniparametric family with an order of convergence of six for any real or complex values of the parameters \(\alpha\), \(\beta\) and \(\gamma\), as long as (27) is satisfied. Therefore, the iterative expression of the new uniparametric family, dependent only on parameter \(\alpha\) and which we will call CMT(\(\alpha\)) family, is defined as  
\[\left\{ \begin{aligned} y_{k}&=x_{k}-\frac{f(x_{k})}{f^ {\prime}(x_{k})},\\ z_{k}&=y_{k}-\frac{f(y_{k})}{2f[x_{k},y_{k}]-f^{\prime }(x_{k})},\\ x_{k+1}&=z_{k}-(\alpha+(1+\alpha)u_{k}+(1-\alpha)v_{ k})\frac{f(z_{k})}{f^{\prime}(x_{k})},\end{aligned} \right.