# An Improved Convergence Analysis of a Multi-Step Method with High-Efficiency Indices
## 3 Local Convergence Analysis  
We will be using the following extra assumptions in our local analysis. * The condition (5) of the lemma holds for \(\mu=\frac{1}{2L_{0}}\). * \(\|\mathcal{G}^{-1}(\AE^{\prime\prime}(s)-\AE^{\prime\prime}(t))\|\leq L_{2}\| s-t\|\) for some \(L_{2}>0\) and \(s,t\in\Omega_{1}\). * \(\|\mathcal{G}^{-1}\AE^{\prime}(s)\|\leq K_{1}\) for some \(K_{1}>0\) and \(s\in\Omega_{1}\). * \(\|\mathcal{G}^{-1}\AE^{\prime\prime}(s)\|\leq K_{2}\) for some \(K_{2}>0\) and \(s\in\Omega_{1}\). We obtained from our semi-local analysis that the solution \(s^{*}\in\bar{U}(s_{0},\lambda)\subset U(s_{0},\frac{1}{2L_{0}})\). Then, by \((a_{2})\), for all \(s\in U(s_{0},\frac{1}{2L_{0}})\), we can get  
\[\|\mathcal{G}^{-1}(\AE^{\prime}(s)-\mathcal{G})\| \leq L_{0}\|s-s_{0}\|\leq L_{0}(\|s-s^{*}\|+\|s^{*}-s_{0}\|)\] \[\leq L_{0}(\|s-s^{*}\|+\frac{1}{2L_{0}})\] \[= \frac{1}{2}+L_{0}\|s-s^{*}\|.\]  
Now, using \(\AE^{\prime}(s)\) is invertible for all \(s\in U(s_{0},\frac{1}{2L_{0}})\), and  
\[\|\AE^{\prime}(s)^{-1}\mathcal{G}\|\leq\frac{2}{1-2L_{0}\|s-s^{*}\|}. \tag{21}\]  
We will use the following inequality in our study. For all \(s,t\in\Omega_{1}\), by MVT, we get  
\[\|\AE^{\prime}(s)^{-1}\AE(t)\| = \AE^{\prime}(s)^{-1}(\AE(t)-\AE(s^{*}))\] \[\leq \|\AE^{\prime}(s)^{-1}\int_{0}^{1}\AE^{\prime}(s^{*}+\tau(t-s^{*} ))d\tau(t-s^{*})\|\] \[\leq \|\AE^{\prime}(s)^{-1}\mathcal{G}\|\int_{0}^{1}\|\mathcal{G}^{-1 }\AE^{\prime}(s^{*}+\tau(t-s^{*}))\|d\tau\|t-s^{*}\|.\]  
Moreover, using assumption \((a_{7})\) and (21), for \(s,t\in U(s^{*},\frac{1}{2L_{0}})\), we obtain  
\[\|\AE^{\prime}(s)^{-1}\AE(t)\|\leq\frac{2K_{1}}{1-2L_{0}\|s-s^{*}\|}\|t-s^{*}\|. \tag{22}\]  
We study the local convergence in the ball \(U(s^{*},r)\), which satisfies  
\[U(s^{*},r)\subset U(s^{*},\frac{1}{2L_{0}})\subset\Omega_{1}. \tag{23}\]_Hence, hereafter we select \(s_{0}\) from \(U(s^{*},r)\)._  
_Graphical representation of (23) is given in Figure 1._  
As a consequence of (23), all the assumptions we have made remain valid in the local convergence ball. So, we can continue our local analysis independently under the same set of assumptions. We need the following theorem for our local convergence analysis. Theorem 2 ([20]): _Let \(F:\Omega\subset X\to Y\) be twice differentiable at the point \(a\), then_  
\[\leq L_{0}(\|s-s^{*}\|+\frac{1}{2L_{0}})\]0  
Proposition 2: _If \(s_{0}\in U(s^{*},\tilde{r})\), where \(\tilde{r}=min\{\frac{1}{L_{1}+2L_{0}},\rho_{1}\}\), \(\rho_{1}\) is the smallest zero of \(h_{1}\) on \([0,\frac{1}{2L_{0}})\),_  
\[\leq L_{0}(\|s-s^{*}\|+\frac{1}{2L_{0}})\]1 _and_ \[\leq L_{0}(\|s-s^{*}\|+\frac{1}{2L_{0}})\]2  
_Then, under assumptions \((a_{1})-(a_{3})\), we have \(t_{0},v_{0}\in U(s,\tilde{r})\) with_  
\[\leq L_{0}(\|s-s^{*}\|+\frac{1}{2L_{0}})\]3 _and_ \[\leq L_{0}(\|s-s^{*}\|+\frac{1}{2L_{0}})\]4  
Proof: Note that \(\phi_{1},h_{1}\) are non-decreasing continuous functions (NDCF) on \([0,\frac{1}{2L_{0}})\), with  
\[\leq L_{0}(\|s-s^{*}\|+\frac{1}{2L_{0}})\]5  
So, by Intermediate Value Theorem (IVT), there exists a smallest \(\rho_{1}\in(0,\frac{1}{2L_{0}})\) such that \(h_{1}(\rho_{1})=0\). \[\leq L_{0}(\|s-s^{*}\|+\frac{1}{2L_{0}})\]6 A visual comparison of the different balls discussed. Note that  
\[\leq L_{0}(\|s-s^{*}\|+\frac{1}{2L_{0}})\]7 \[\leq L_{0}(\|s-s^{*}\|+\frac{1}{2L_{0}})\]8  
So, by the assumption \((a_{3})\) and (21), we obtain  
\[\leq L_{0}(\|s-s^{*}\|+\frac{1}{2L_{0}})\]9 \[= \frac{1}{2}+L_{0}\|s-s^{*}\|.\]0  
Thus, the iterate \(t_{0}\in U(s^{*},\tilde{r})\). Also, by (22) and the fact that \(\phi(t)t<1\), \(\forall t\in(0,\rho_{1})\), we have  
\[= \frac{1}{2}+L_{0}\|s-s^{*}\|.\]1 \[= \frac{1}{2}+L_{0}\|s-s^{*}\|.\]2 \[= \frac{1}{2}+L_{0}\|s-s^{*}\|.\]3 \[= \frac{1}{2}+L_{0}\|s-s^{*}\|.\]4 \[= \frac{1}{2}+L_{0}\|s-s^{*}\|.\]5  
Hence, the iterate \(v_{0}\in U(s^{*},\tilde{r})\).