# Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations
## 4 Numerical Results
### First Experiment: Efficiency Analysis of CMT (\(\alpha\)) Family
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Function** & **Method** & \(|\mathbf{x_{k+1}-x_{k}}|\) & \(|\mathbf{f(x_{k+1})}|\) & **iter** & **ACOC** & **tcpu** \\ \hline \(f_{1}\) & CMT(1) & \(7.6395\times 10^{-19}\) & \(1.8769\times 10^{-110}\) & 3 & 5.5148 & 0.1257 \\ \(x_{0}=-1.6\) & Newton & \(3.2063\times 10^{-84}\) & \(7.2243\times 10^{-168}\) & 8 & 2 & 0.1225 \\  & Ostrowski & \(3.6277\times 10^{-39}\) & \(2.1775\times 10^{-155}\) & 4 & 3.9988 & 0.1036 \\  & ABA & \(6.3941\times 10^{-19}\) & \(5.2542\times 10^{-111}\) & 3 & 5.5472 & 0.1201 \\  & CH & \(3.9619\times 10^{-19}\) & \(3.095\times 10^{-112}\) & 3 & 5.5336 & 0.1173 \\  & AHR & \(6.9779\times 10^{-86}\) & 0 & 4 & 5.9989 & 0.1381 \\ \hline \(f_{2}\) & CMT(1) & \(1.1915\times 10^{-19}\) & \(3.2336\times 10^{-114}\) & 4 & 6.0717 & 0.2913 \\ \(x_{0}=-0.4\) & Newton & \(6.977\times 10^{-101}\) & \(9.2573\times 10^{-201}\) & 10 & 2 & 0.2747 \\  & Ostrowski & \(3.6009\times 10^{-28}\) & \(5.8295\times 10^{-111}\) & 4 & 3.9993 & 0.1899 \\  & ABA & \(6.593\times 10^{-46}\) & 0 & 4 & 6.0055 & 0.4278 \\  & CH & \(4.0133\times 10^{-50}\) & \(6.8135\times 10^{-208}\) & 5 & 5.9951 & 0.5636 \\  & AHR & \(1.4561\times 10^{-73}\) & 0 & 10 & 5.9991 & 0.7038 \\ \hline \(f_{3}\) & CMT(1) & \(5.868\times 10^{-64}\) & 0 & 7 & 5.9957 & 0.4654 \\ \(x_{0}=0.4\) & Newton & \(3.2665\times 10^{-83}\) & \(8.6382\times 10^{-165}\) & 10 & 2 & 0.2818 \\  & Ostrowski & \(1.3665\times 10^{-51}\) & \(5.077\times 10^{-204}\) & 5 & 3.9999 & 0.1682 \\  & ABA & \(2.5625\times 10^{-27}\) & \(4.3729\times 10^{-160}\) & 5 & 5.8933 & 0.224 \\  & CH & \(2.4971\times 10^{-24}\) & \(4.0266\times 10^{-142}\) & 9 & 5.8498 & 0.4374 \\  & AHR & \(2.1589\times 10^{-36}\) & 0 & 12 & 5.9521 & 0.5912 \\ \hline \(f_{4}\) & CMT(1) & \(1.2572\times 10^{-32}\) & \(3.2096\times 10^{-195}\) & 3 & 5.717 & 0.6075 \\ \(x_{0}=1.3\) & Newton & \(7.2803\times 10^{-95}\) & \(1.2821\times 10^{-189}\) & 7 & 2 & 0.4947 \\  & Ostrowski & \(1.0395\times 10^{-64}\) & \(1.5574\times 10^{-207}\) & 4 & 4 & 0.535 \\  & ABA & \(4.7735\times 10^{-26}\) & \(8.9685\times 10^{-156}\) & 3 & 5.9419 & 0.598 \\  & CH & \(1.6112\times 10^{-32}\) & \(1.7919\times 10^{-194}\) & 3 & 5.6961 & 0.6046 \\  & AHR & \(3.0816\times 10^{-22}\) & \(1.1423\times 10^{-131}\) & 3 & 5.6812 & 0.4497 \\ \hline \(f_{5}\) & CMT(1) & \(2.5535\times 10^{-53}\) & \(6.4242\times 10^{-207}\) & 6 & 5.9132 & 1.2222 \\ \(x_{0}=-1.9\) & Newton & \(3.4167\times 10^{-84}\) & \(8.1562\times 10^{-167}\) & 8 & 2 & 0.6295 \\  & Ostrowski & \(4.1408\times 10^{-38}\) & \(2.4627\times 10^{-142}\) & 4 & 4.0146 & 0.5521 \\  & ABA & \(5.637\times 10^{-65}\) & \(1.9467\times 10^{-208}\) & 5 & 6.0107 & 0.9561 \\  & CH & \(1.0828\times 10^{-43}\) & \(1.0707\times 10^{-207}\) & 6 & 6.2212 & 1.1314 \\  & AHR & \(5.6988\times 10^{-26}\) & \(4.6285\times 10^{-106}\) & 4 & 5.7855 & 0.5939 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical performance of iterative methods in nonlinear equations for \(x_{0}\)**close to \(\xi\)**. On the other hand, from the results shown in Tables 7-9, we see that the methods associated with \(\alpha=-300,200,400\) do not always converge to the solution, confirming the conclusions obtained in the dynamical analysis. The convergence highly depends on the initial estimation and the nonlinear test function used. Thus, for estimations close to the root, these methods do not converge to the solution in up to 2 test functions. Moreover, for estimations far and very far from the root, they do not converge to the solution even for any function. Consequently, we conclude that the methods for \(\alpha=-1,\ 0,\ 1\) are stable, have the lowest processing times (tcpu), and always converge to the solution for any initial estimate and nonlinear test function used. The methods for \(\alpha=-300,\ 200,\ 400\) are unstable, chaotic, have the highest tcpu, and tend not to converge to the solution according to the initial estimate and the nonlinear test function used.