# Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations
## 5 Conclusions  
In this paper, a new family of iterative methods was designed to solve nonlinear equations from Ostrowski scheme, adding a Newton step with a \"frozen\" derivative and using a divided difference operator. This family, named CMT \((\alpha,\beta,\gamma)\), has a three-step iterative expression and three arbitrary parameters which can take any real or complex value. In the convergence analysis of the new family, we obtained an order of convergence of four just like the order of the Ostrowski method. However, we managed to speed-up the convergence to six by setting the parameters \(\beta\) and \(\gamma\) as a function of \(\alpha\), resulting in an uniparametric CMT \((\alpha)\) family. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & \(x_{0}\) & \(|x_{k+1}-x_{k}|\) & \(|f(x_{k+1})|\) & **iter** & **ACOC** & **tcpu** \\ \hline \multicolumn{6}{c}{Close to \(\xi\)} \\ \(f_{1}\) & \(-1.6\) & \(2.1496\times 10^{-56}\) & \(0\) & \(4\) & \(5.9921\) & \(0.1499\) \\ \(f_{2}\) & \(-0.4\) & nc & nc & nc & nc & nc \\ \(f_{3}\) & \(0.4\) & nc & nc & nc & nc & nc \\ \(f_{4}\) & \(1.3\) & \(4.0045\times 10^{-33}\) & \(1.6998\times 10^{-196}\) & \(3\) & \(5.3325\) & \(0.6325\) \\ \(f_{5}\) & \(-1.9\) & \(1.3149\times 10^{-70}\) & \(9.6363\times 10^{-207}\) & \(4\) & \(6.0496\) & \(0.8213\) \\ \hline \multicolumn{6}{c}{Far from \(\xi\)} \\ \(f_{1}\) & \(-6\) & \(6.1599\times 10^{-40}\) & \(3.8934\times 10^{-208}\) & \(7\) & \(5.9673\) & \(0.2711\) \\ \(f_{2}\) & \(-6\) & nc & nc & nc & nc & nc \\ \(f_{3}\) & \(-14\) & nc & nc & nc & nc & nc \\ \(f_{4}\) & \(-23\) & \(4.3946\times 10^{-33}\) & \(2.9689\times 10^{-196}\) & \(7\) & \(5.339\) & \(1.4742\) \\ \(f_{5}\) & \(-9\) & \(8.369\times 10^{-63}\) & \(2.9162\times 10^{-205}\) & \(11\) & \(5.964\) & \(2.0915\) \\ \hline \multicolumn{6}{c}{Very far from \(\xi\)} \\ \(f_{1}\) & \(-60\) & \(1.9877\times 10^{-20}\) & \(1.8239\times 10^{-118}\) & \(14\) & \(5.7565\) & \(0.5598\) \\ \(f_{2}\) & \(-60\) & nc & nc & nc & nc & nc \\ \(f_{3}\) & \(-140\) & nc & nc & nc & nc & nc \\ \(f_{4}\) & \(-230\) & \(2.7541\times 10^{-49}\) & \(1.5574\times 10^{-207}\) & \(15\) & \(5.9586\) & \(3.1228\) \\ \(f_{5}\) & \(-90\) & \(7.8278\times 10^{-51}\) & \(9.6363\times 10^{-207}\) & \(15\) & \(6.1663\) & \(3.2771\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Numerical performance of CMT (200) method in nonlinear equations. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & \(x_{0}\) & \(|x_{k+1}-x_{k}|\) & \(|f(x_{k+1})|\) & **iter** & **ACOC** & **tcpu** \\ \hline \multicolumn{6}{c}{Close to \(\xi\)} \\ \(f_{1}\) & \(-1.6\) & \(2.9103\times 10^{-44}\) & \(0\) & \(4\) & \(5.9805\) & \(0.1439\) \\ \(f_{2}\) & \(-0.4\) & nc & nc & nc & nc & nc \\ \(f_{3}\) & \(0.4\) & nc & nc & nc & nc & nc \\ \(f_{4}\) & \(1.3\) & \(1.139\times 10^{-35}\) & \(1.5574\times 10^{-207}\) & \(3\) & \(5.2494\) & \(0.6218\) \\ \(f_{5}\) & \(-1.9\) & \(5.8131\times 10^{-53}\) & \(3.1147\times 10^{-207}\) & \(4\) & \(5.754\) & \(0.8023\) \\ \hline \multicolumn{6}{c}{Far from \(\xi\)} \\ \(f_{1}\) & \(-6\) & nc & nc & nc & nc & nc \\ \(f_{2}\) & \(-6\) & nc & nc & nc & nc & nc \\ \(f_{3}\) & \(-140\) & nc & nc & nc & nc & nc \\ \(f_{4}\) & \(-230\) & nc & nc & nc & nc & nc \\ \(f_{5}\) & \(-90\) & nc & nc & nc & nc & nc \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Numerical performance of CMT (400) method in nonlinear equations. In the dynamical study, we constructed parameters spaces of the free critical points of the rational operator associated with the uniparametric family. These parameter spaces allowed us to understand the performance of the different members of the family, helping us to choose stable (for \(\alpha=-1,\ 0,\ 1,...\)) and unstable (for \(\alpha=-300,\ 200,\ 400,...\)) methods. Furthermore, we generated dynamical planes to show the behavior of these particular methods. From numerical results, the order of convergence is verified by the ACOC, which is close to 6.