# Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative
## 2 Local Convergence: One Dimensional Case
Moreover, the following estimates hold_  
\[h_{3}(t)=g_{3}(t)-1.\]7  
\[h_{3}(t)=g_{3}(t)-1.\]8  
_and_  
\[h_{3}(t)=g_{3}(t)-1.\]9  
_where the \"\(g\)\" functions are defined by previously. Furthermore, for \(T\in\left[r,\ \frac{2}{\bar{L_{0}}}\right)\), the limit point \(x^{*}\) is the only solution of equation \(F(x)=0\) in \(\bar{U}(x^{*},\ T)\cap\mathbb{D}_{0}\)._  
Then, using the last substep of method Equation (2) for \(n=0\), Equations (4), (10), (13)-(15), (19), (20) (for \(n=0\)), (23), (26) and (29), we obtain that  
\[ \begin{split}\|x_{1}-x^{*}\|&\leq\|z_{0}-x^{*}\|+\left\| F^{\prime}(x_{0})^{-1}F^{\prime}(x^{*})\right\|\left[\ \left\|F^{\prime}(x^{*})^{-1}\big{(}[y_{0},\ x_{0};\ F]-F^{\prime}(x^{*}) \big{)}\right\|\right.\\ &\quad+\left\|F^{\prime}(x^{*})^{-1}\big{(}F^{\prime}(x_{0})-F^{ \prime}(x^{*})\big{)}\right\|+\left\|F^{\prime}(x^{*})^{-1}[z_{0},\ x_{0};\ F]\right\| \right]\\ &\times\left\|\big{(}2[z_{0},\ y_{0};\ F]-[z_{0},\ x_{0};\ F] \big{)}^{-1}\,F^{\prime}(x^{*})\right\|\left\|F^{\prime}(x^{*})^{-1}F(z_{0}) \right\|\\ &\leq\|z_{0}-x^{*}\|+\frac{\big{(}K+L_{0}\|x_{0}-x^{*}\|+K_{0}\| y_{0}-x^{*}\|+K_{1}\|x_{0}-x^{*}\|\big{)}M\|z_{0}-x^{*}\|}{(1-L_{0}\|x_{0}-x^{*} \|)\,(1-q(\|x_{0}-x^{*}\|))}\\ &\leq\left(1+\frac{\big{(}K+(L_{0}+K_{1})\|x_{0}-x^{*}\|+K_{0}g _{1}(\|x_{0}-x^{*}\|)\|x_{0}-x^{*}\|\big{)}M}{(1-L_{0}\|x_{0}-x^{*}\|)\,(1-q( \|x_{0}-x^{*}\|))}\right)\|z_{0}-x^{*}\|\\ &\leq g_{3}(\|x_{0}-x^{*}\|)\|x_{0}-x^{*}\|<\|x_{0}-x^{*}\|<r, \end{split}  \tag{30}\]  
which shows Equation (21) for \(n=0\) and \(x_{1}\in U(x^{*},\ r)\). By simply replacing \(x_{0}\), \(y_{0},\ z_{0}\) by \(x_{n},y_{n}\), \(z_{n}\) in the preceding estimates we arrive at Equations (19)-(21) for all \(n=0,\ 1,\ 2,\ \ldots\). Using the monotonicity of \(g_{3}\) on the interval \([0,\ r]\), \(x_{0}\in U(x^{*},\ r)\) and (21), we have with \(c=g_{3}(\|x_{0}-x^{*}\|)\in[0,\ 1)\)  
\[\|x_{n+1}-x^{*}\|\leq c\|x_{n}-x^{*}\|\leq\cdots\leq c^{n}\|x_{0}-x^{*}\|, \tag{31}\]  
where \(r<1\). This yields \(\lim\limits_{n\to\infty}x_{n}=x^{*}\) and \(x_{n+1}\in U(x^{*},\ r)\). Finally, to show the uniqueness part, let \(y^{*}\in\bar{U}(x^{*},T)\) be such that \(F(y^{*})=0\). Set \(Q=\int_{0}^{1}F^{\prime}\left(x^{*}+\theta(y^{*}-x^{*})\right)d\theta\). Then, using Equation (13), we get that  
\[\left\|F^{\prime}(x^{*})^{-1}(Q-F^{\prime}(x^{*}))\right\|\leq L_{0}\int_{0}^ {1}\theta\|x^{*}-y^{*}\|d\theta\leq\frac{L_{0}}{2}T<1. \tag{32}\]  
Hence, \(Q^{-1}\in L(\mathbb{Y},\ \mathbb{X})\). Then, in view of the identity \(F(y^{*})-F(x^{*})=Q(y^{*}-x^{*})\), we conclude that \(x^{*}=y^{*}\). It is worth noticing that in view of Equations (12), (21), (31) and the definition of functions \(g_{i},\ i=1,2,3\) the convergence order of method Equation (2) is at least \(\lambda\). **Remark 1**  
(a) In view of Equation (13) and the estimate  
\[ \begin{array}{rcl}\left\|F^{\prime}(x^{*})^{-1}F^{\prime}(x)\right\|&=&\left\| F^{\prime}(x^{*})^{-1}(F^{\prime}(x)-F^{\prime}(x^{*}))+F^{\prime}(x^{*})\right\| \\ &\leq&1+\left\|F^{\prime}(x^{*})^{-1}(F^{\prime}(x)-F^{\prime}(x^{*}))\right\| \\ &\leq&1+L_{0}\|x-x^{*}\|\end{array} \]  
condition Equation (17) can be dropped and \(M\) can be replaced by  
\[M=M(t)=1+L_{0}t\]  
or \(M=2\), since \(t\in[0,\frac{1}{L_{0}})\). (b) The results obtained here can be used for operators \(F\) satisfying the autonomous differential equation [4,5] of the form  
\[F^{\prime}(x)=P(F(x)),\]where \(P\) is a known continuous operator. Since \(F^{\prime}(x^{*})=P(F(x^{*}))=P(0)\), we can apply the results without actually knowing the solution \(x^{*}\). As an example, let \(F(x)=e^{x}+2\). Then, we can choose \(P(x)=x-2\). 3. The radius \(r_{1}\) was shown in [4; 5] to be the convergence radius for [PERSON]'s method under conditions (11) and (12). It follows from Equation (4) and the definition of \(r_{1}\) that the convergence radius \(r\) of the method Equation (2) cannot be larger than the convergence radius \(r_{1}\) of the second order [PERSON]'s method.