# Local Convergence Study for an Iterative Scheme with a High Order of Convergence
## 2 Local Convergence Analysis
### Local Convergence Analysis Using Condition (\(C_{4}\))
\begin{split}&\left\|\mathcal{T}^{\prime}(x^{*})^{-1}\mathcal{T}^{ \prime}(x^{*}+t(x_{0}-x^{*}))\right\|\leq M\\ &\left\|\mathcal{T}^{\prime}(x^{*})^{-1}\mathcal{T}(x_{0}) \right\|\leq M\|x_{0}-x^{*}\|\end{split}  \tag{15}\]  
The proof of this lemma can be accomplished by using the definition of \(\overline{r}\) and \(C_{4}\). Proof of Lemma 2.: We have that:  
\[\mathcal{T}(x_{0})=\mathcal{T}(x_{0})-\mathcal{T}(x^{*})=\int_{x^{*}}^{x_{0}} \mathcal{T}^{\prime}(z)dz=\int_{0}^{1}\mathcal{T}^{\prime}(x^{*}+t(x_{0}-x^{ *}))(x_{0}-x^{*})dt \tag{16}\]  
Notice that \(\|x^{*}+t(x_{0}-x^{*})-x^{*}\|=t\|x_{0}-x^{*}\|\leq\|x_{0}-x^{*}\|\leq\overline {r}\). Then, \(\|x^{*}+t(x_{0}-x^{*})\|\in B(x^{*},\overline{r})\). Using (16) and (C\({}_{4}\)), we obtain  
\[\left\|\mathcal{T}^{\prime}(x^{*})^{-1}\mathcal{T}^{\prime}(x^{*}+t(x_{0}-x^ {*}))\right\|\leq M.\]  
Consequently:  
\[ \begin{split}\left\|\mathcal{T}^{\prime}(x^{*})^{-1}\mathcal{T}( x_{0})\right\|&=\left\|\mathcal{T}^{\prime}(x^{*})^{-1}\int_{0}^{1} \mathcal{T}^{\prime}(x^{*}+t(x_{0}-x^{*}))(x_{0}-x^{*})dt\right\|\\ &\leq\int_{0}^{1}\left\|\mathcal{T}^{\prime}(x^{*})^{-1} \mathcal{T}^{\prime}(x^{*}+t(x_{0}-x^{*}))(x_{0}-x^{*})\right\|dt\\ &\leq\int_{0}^{1}M\|x_{0}-x^{*}\|dt=M\|x_{0}-x^{*}\|t\mid_{0}^{1 }\\ &=M\|x_{0}-x^{*}\|.\end{split} \]  
We will now present the local convergence result for scheme (1), given the (C\({}_{1}\))-(C\({}_{4}\)) conditions. **Theorem 2**. : _Let \(\mathcal{T}:D\subseteq X\to Y\) be a differentiable Frechet operator, \(L_{0}>0,L>0\) and \(M>1\). Let \(x^{*}\in D\) be such that conditions (C\({}_{1}\))-(C\({}_{4}\)) are satisfied for all \(x,y\in D\), and moreover, it is verified that \(M<3\) and \(\left|\frac{3}{8}+\alpha\right|M+\left|\alpha\right|M+\left|\frac{\alpha}{3} \right|M^{2}+\left|\frac{3}{8}-\frac{\alpha}{3}\right|M^{2}<1\). Then, the sequence \(\{x_{k}\}\) generated by (1) for \(x_{0}\in B(x^{*},\overline{r})-\{x^{*}\}\) is well defined for \(k=0,1,2,\dots\) remains in \(B(x^{*},\overline{r})\) and converges to \(x^{*}\). In addition, the estimates provided below hold the following conditions for \(k=0,1,2,\dots\):_  
\[ \begin{split}\|y_{k}-x^{*}\|&\leq\overline{g_{1}}( \|x_{k}-x^{*}\|)\|x_{k}-x^{*}\|\leq\|x_{k}-x^{*}\|\leq\overline{r},\\ \|x_{k+1}-x^{*}\|&\leq\overline{g_{2}}(\|x_{k}-x^{*} \|)\|x_{k}-x^{*}\|\leq\|x_{k}-x^{*}\|\leq\overline{r},\end{split} \]  
_where the \(\overline{g}\) functions are defined before the introduction of Theorem 2. Furthermore, if \(\overline{R}\in\left[\overline{r},\frac{2}{L_{0}}\right]\) exists such that \(\overline{B}(x^{*},\overline{R})\subseteq D\), then the limit point \(x^{*}\) stands as the unique solution to the equation \(\mathcal{T}(x)=0\) in \(\overline{B}(x^{*},\overline{R})\)._  
Proof of Theorem 2.: We do not include the proof of the preceding theorem as it is nearly the same as Theorem 1. The uniqueness proof is shown in Theorem 1. Next, we perform a comparative study of the Efficiency Index \((EI)\) and the Computational Efficiency Index \((CEI)\) between schemes (1), [PERSON] [13], [PERSON] [14], and [PERSON] et al. [15].