# Dynamics and Stability on a Family of Optimal Fourth-Order Iterative Methods
## 7 Conclusions  
In our manuscript, we found a large, optimal, and general kind of fourth-order scheme that is free of second derivatives. The dynamical analysis that was performed on the proposed class of methods allowed us to choose family members that were particularly stable, discarding the ones with unstable performance, where strange fixed points and some periodical orbits were present. Lastly, we refer to Table 2 to show that the suggested family is equal to or more competitive than the classical method, the recognized and efficient methods of [PERSON] and [PERSON], and other recognized and efficient methods found in the literature. The dynamic study shows us that there are values of the parameter \(\beta\) for which we can obtain highly efficient iterative families of methods; some of the values that we can use are \(\beta=0\), \(\beta=1\), \(\beta=10\), and \(\beta=100\).  
This behavior is justified in the dynamic planes given in Figure 5a-d, since there is global convergence to the roots of a second-degree polynomial.  
In addition, if we use parameter values corresponding to regions of instability, one of the consequences of using these schemes is that these do not converge to the root, and if they do converge, they would do so with a large number of iterations, that is, these members are unreliable. This fact can be confirmed in Figure 4a-d.  
The results shown in Tables 2 and 3 are satisfactory from the point of view of numerical calculations. It can be observed that the number of iterations, computational time, and theoretical convergence order reached by our family, in comparison with the known methods of [PERSON] and the BCMT, are satisfactory, as we expected, since we took one of the values of the \(beta\) parameter for which it was theoretically shown that we would have efficient iterative methods. In addition, another piece of evidence that shows the degree of accuracy with which our method converges to the root is the computation of the absolute errors \(|x_{n+1}-x_{n}|\) and \(|f(x_{n+1})|\), which are shown in Table 3.  
**Author Contributions:** Conceptualization, A.C. and J.R.T.; software, M.A.L.S.; validation, A.C. and J.R.T.; formal analysis, J.R.T.; investigation, A.C. and M.A.L.S.; writing--original draft preparation, M.A.L.S.; writing--review and editing, A.C. and J.R.T.; visualization, M.A.L.S.; supervision, A.C. and J.R.T. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by PGC2018-095896-B-C22 and by MCIN/AEI/10.13039/ 5011000113033 as part of \"ERDF: A way to making Europe\".  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Acknowledgments:** The authors would like to thank the anonymous reviewers for their comments and suggestions, as they have improved the final version of this manuscript.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \(f(x)\) & & \(|x_{n+1}-x_{n}|\) & & & \(|f(x_{n+1})|\) & \\ \hline  & **M1** & **M2** & **M3** & **M1** & **M2** & **M3** \\ \hline \(f_{1}(x)\) & \(7.3\times 10^{-16}\) & \(2.59\times 10^{-16}\) & \(1.67\times 10^{-16}\) & \(1.16\times 10^{-62}\) & \(2.87\times 10^{-64}\) & \(2.58\times 10^{-65}\) \\ \(f_{2}(x)\) & \(1.16\times 10^{-27}\) & \(1.35\times 10^{-13}\) & \(2.31\times 10^{-19}\) & \(2.19\times 10^{-207}\) & \(2.17\times 10^{-50}\) & \(3.84\times 10^{-74}\) \\ \(f_{3}(x)\) & \(5.69\times 10^{-10}\) & \(9.24\times 10^{-9}\) & \(1.33\times 10^{-10}\) & \(5.64\times 10^{-27}\) & \(8.89\times 10^{-32}\) & \(1.11\times 10^{-39}\) \\ \(f_{4}(x)\) & \(3.7\times 10^{-19}\) & \(1.3\times 10^{-25}\) & \(2.18\times 10^{-28}\) & \(2.22\times 10^{-73}\) & \(7.57\times 10^{-99}\) & \(1.79\times 10^{-110}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Benchmark comparison of fourth-order methods in relation to the absolute error \(|x_{n+1}-x_{n}|\) and \(|f(x_{n+1})|\).