# Local Convergence Study for an Iterative Scheme with a High Order of Convergence
## 5 Conclusions  
The current paper presents a local convergence study of a class family of iterative schemes within Banach spaces, provided that the Frechet derivative satisfies the Lipschitz continuity condition. For the convergence balls, the radii have been obtained, according to the existence and uniqueness Theorem. In the examples shown, scheme (1) gives smaller radii without using \(M\). All schemes have the same [PERSON]'s Efficiency Index, since this index focuses exclusively on the number of functional evaluations and the order of convergence. Scheme (1) has better behavior for \(\alpha=0\) (in terms of Computational Efficiency) than all the other methods analyzed. Furthermore, [PERSON] and \(M1\) for \(\alpha=\frac{9}{8}\) are second in efficiency. For the Computational Efficiency Index was introduced a correction factor \(\mu\), which transformed the number of evaluations into a number of operations. The value of this parameter can be tested computationally to establish a relationship between evaluating a function and performing arithmetic operations. Nevertheless, the key is to use the same value as in other studies to ensure a proper comparison. Future works related to this line of research and based on these results are focused on semilocal convergence analysis using majorizing sequences. This study extends the scope of local convergence by not only establishing conditions under which an initial approximation leads to a solution, but also providing the domain of existence of that solution, thus being valid for proving the existence of a solution to a problem for which prior information was not known. This analysis is particularly valuable for demonstrating the existence of solutions and ensuring the reliability of iterative algorithms in functional analysis, optimization, and numerical applications. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & \(\tilde{\pi}\) & \(\left\|x_{\mathbf{k+1}}-x_{\mathbf{k}}\right\|\) & \(\mathcal{T}(x_{\mathbf{k+1}})\) & **Iter** & **ACOC** & **e-Time** \\ \hline \multirow{4}{*}{\(M1_{(\alpha=0)}\)} & \multicolumn{4}{c}{\(x_{0}=[0.5\ \ 0.5\ 0.5]^{T}\)} \\  & \(\left[1.18\times 10^{-40},0,0\right]^{T}\) & \(1.17\times 10^{-31}\) & \(1.17\times 10^{-31}\) & 4 & 3.9754 & 0.33349 \\ \(M1_{(\alpha=\frac{1}{8})}\) & nc & \(1.13\times 10^{17}\) & \(9.22\times 10^{51}\) & 50 & 1.00000 & 3.68628 \\ Jaiswal & \(\left[4.43\times 10^{-41},-7.24\times 10^{-71},0\right]^{T}\) & \(8.49\times 10^{-32}\) & \(8.49\times 10^{-32}\) & 4 & 3.9972 & 0.32919 \\ CHJ & \(\left[2.7\times 10^{-43},-1.19\times 10^{-66},0\right]^{T}\) & \(1.62\times 10^{-27}\) & \(1.62\times 10^{-27}\) & 4 & 3.9454 & 0.35557 \\ Sharma & nc & \(1.13\times 10^{17}\) & \(9.22\times 10^{51}\) & 50 & 1.00000 & 3.60789 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical performance of iterative schemes for a nonlinear system of equations. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & \(\tilde{\pi}\) & \(\left\|x_{\mathbf{k+1}}-x_{\mathbf{k}}\right\|\) & \(\mathcal{T}(x_{\mathbf{k+1}})\) & **Iter** & **ACOC** & **e-Time** \\ \hline \multirow{4}{*}{\(M1_{(\alpha=0)}\)} & \multicolumn{4}{c}{\(x_{0}=[1,1,\ldots,1]^{T}\)} \\  & \(\left[0.514933,\,0.514933,\ldots,0.514933\right]^{T}\) & \(1.18\times 10^{-11}\) & \(3.21\times 10^{-11}\) & 3 & 3.6538 & 1.78310 \\ \(M1_{(\alpha=\frac{1}{8})}\) & nc & \(3.36\times 10^{30}\) & \(1.03\times 10^{31}\) & 50 & - & 33.70610 \\ Jaiswal & \(\left[0.514933,\,0.514933,\ldots,0.514933\right]^{T}\) & \(1.23\times 10^{-11}\) & \(3.35\times 10^{-11}\) & 3 & 3.6476 & 1.80741 \\ CHJ & \(\left[0.514933,\,0.514933,\ldots,0.514933\right]^{T}\) & \(1.73\times 10^{-11}\) & \(4.69\times 10^{-11}\) & 3 & 3.5972 & 1.78408 \\ [PERSON] & nc & \(3.36\times 10^{30}\) & \(1.03\times 10^{31}\) & 50 & - & 40.46302 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numerical performance of iterative schemes for a nonlinear system of equations. **Author Contributions:** Conceptualization, A.L. and E.M.; methodology, A.L. and E.M.; software, A.L.