## 3 Convergence Analysis
\[\|G(x)-G(\alpha)-G^{\prime}(\alpha)(x-\alpha)\|\] \[=\|P(x)-Q(x)R(x)-\alpha-(P^{\prime}(\alpha)-Q(\alpha)R^{\prime}( \alpha))(x-\alpha)\|\] \[F^{\prime}(\alpha+h)=F^{\prime}(\alpha)\left[I+\sum_{q=2}^{p-1}qC_{q}h^{q-1} \right]+O(h^{p}) \tag{9}\]0  
Since \(P(x)\) is differentiable in \(\alpha\) and \(P(\alpha)=\alpha\), we can assume that \(\delta\) was chosen sufficiently small such that  
\[F^{\prime}(\alpha+h)=F^{\prime}(\alpha)\left[I+\sum_{q=2}^{p-1}qC_{q}h^{q-1} \right]+O(h^{p}) \tag{9}\]1  
for all \(x\in S\) with \(\epsilon>0\) depending on \(\delta\) and \(\epsilon=0\) in case \(P(x)=x\). Since \(P\), \(Q\) and \(R\) are continuously differentiable functions, then \(Q^{\prime}\), \(R^{\prime}\) and \(R^{\prime\prime}\) are bounded:  
\[F^{\prime}(\alpha+h)=F^{\prime}(\alpha)\left[I+\sum_{q=2}^{p-1}qC_{q}h^{q-1} \right]+O(h^{p}) \tag{9}\]2  
Now by mean value theorem for integrals  
\[F^{\prime}(\alpha+h)=F^{\prime}(\alpha)\left[I+\sum_{q=2}^{p-1}qC_{q}h^{q-1} \right]+O(h^{p}) \tag{9}\]3  
and  
\[F^{\prime}(\alpha+h)=F^{\prime}(\alpha)\left[I+\sum_{q=2}^{p-1}qC_{q}h^{q-1} \right]+O(h^{p}) \tag{9}\]4  
so that  
\[F^{\prime}(\alpha+h)=F^{\prime}(\alpha)\left[I+\sum_{q=2}^{p-1}qC_{q}h^{q-1} \right]+O(h^{p}) \tag{9}\]5 \[F^{\prime}(\alpha+h)=F^{\prime}(\alpha)\left[I+\sum_{q=2}^{p-1}qC_{q}h^{q-1} \right]+O(h^{p}) \tag{9}\]6 \[F^{\prime}(\alpha+h)=F^{\prime}(\alpha)\left[I+\sum_{q=2}^{p-1}qC_{q}h^{q-1} \right]+O(h^{p}) \tag{9}\]7 \[F^{\prime}(\alpha+h)=F^{\prime}(\alpha)\left[I+\sum_{q=2}^{p-1}qC_{q}h^{q-1} \right]+O(h^{p}) \tag{9}\]8 \[F^{\prime}(\alpha+h)=F^{\prime}(\alpha)\left[I+\sum_{q=2}^{p-1}qC_{q}h^{q-1} \right]+O(h^{p}) \tag{9}\]9\[x^{(k+1)}=G(x^{(k)}),\,k=0,1,... \tag{10}\]0 \[x^{(k+1)}=G(x^{(k)}),\,k=0,1,... \tag{10}\]1 \[x^{(k+1)}=G(x^{(k)}),\,k=0,1,... \tag{10}\]2 \[x^{(k+1)}=G(x^{(k)}),\,k=0,1,... \tag{10}\]3  
Combining, we have  
\[x^{(k+1)}=G(x^{(k)}),\,k=0,1,... \tag{10}\]4  
which shows that \(G(x)\) is differentiable in \(\alpha\) since \(\delta\) and \(\epsilon\) are arbitrary and \(\|Q(\alpha)\|\), \(K_{1}\), \(K_{2}\) and \(K_{3}\) are constants. Thus \(G^{\prime}(\alpha)=P^{\prime}(\alpha)-Q(\alpha)R^{\prime}(\alpha)\). **Theorem 3**. : _Let \(F:D\subseteq\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\) be sufficiently Frechet differentiable at each point of an open convex neighborhood \(D\) of \(\alpha\in\mathbb{R}^{n}\) that is a solution of the system \(F(x)=0\). Let us suppose that \(x\in S=\overline{S}(\alpha,\delta)\) and \(F^{\prime}(x)\) is continuous and nonsingular in \(\alpha\), and \(x^{(0)}\) is close enough to \(\alpha\). Then \(\alpha\) is a point of attraction of the sequence \(\{x^{(k)}\}\) obtained using the iterative expression Equation (6). Furthermore, the sequence converges to \(\alpha\) with order \(4\), where the error equation obtained is_  
\[x^{(k+1)}=G(x^{(k)}),\,k=0,1,... \tag{10}\]5  
**Proof:** We first show that \(\alpha\) is a point of attraction using Theorem 2. In this case,  
\[x^{(k+1)}=G(x^{(k)}),\,k=0,1,... \tag{10}\]6  
Now, since \(F(\alpha)=0\), we have  
\[x^{(k+1)}=G(x^{(k)}),\,k=0,1,... \tag{10}\]7 \[x^{(k+1)}=G(x^{(k)}),\,k=0,1,... \tag{10}\]8 \[x^{(k+1)}=G(x^{(k)}),\,k=0,1,... \tag{10}\]9 \[S(\alpha)=\{x\in\mathbb{R}^{n}\Big{|}\left\|x-\alpha\right\|<\delta\},\; \delta>0,\]0 \[S(\alpha)=\{x\in\mathbb{R}^{n}\Big{|}\left\|x-\alpha\right\|<\delta\},\; \delta>0,\]1 \[S(\alpha)=\{x\in\mathbb{R}^{n}\Big{|}\left\|x-\alpha\right\|<\delta\},\; \delta>0,\]2 \[S(\alpha)=\{x\in\mathbb{R}^{n}\Big{|}\left\|x-\alpha\right\|<\delta\},\; \delta>0,\]3  
so that \(\rho(G^{\prime}(\alpha))=0<1\) and by [PERSON]'s theorem, \(\alpha\) is a point of attraction of Equation (6). We next establish the fourth order convergence of this method. From Equations (8) and (9) we obtain  
\[S(\alpha)=\{x\in\mathbb{R}^{n}\Big{|}\left\|x-\alpha\right\|<\delta\},\; \delta>0,\]4  
Also,  
\[