## 3 Convergence Analysis  
The main theorem is going to be demonstrated by means of the n-dimensional Taylor expansion of the functions involved. In the following, we use certain notations and results found in [10]:  
Let \(F:D\subseteq\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\) be sufficiently Frechet differentiable in \(D\). Suppose the \(q\)th derivative of \(F\) at \(u\ \in\mathbb{R}^{n}\), \(q\geq 1\), is the \(q\)-linear function \(F^{(q)}(u):\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}\longrightarrow \mathbb{R}^{n}\) such that \(F^{(q)}(u)(v_{1},\ldots,v_{q})\ \in\ R^{n}\). Given \(\alpha+h\in\mathbb{R}^{n}\), which lies in a neighborhood of a solution \(\alpha\) of the nonlinear system \(F(x)=0\), [PERSON]'s expansion can be applied (assuming Jacobian matrix \(F^{\prime}(\alpha)\) is nonsingular) to obtain  
\[F(\alpha+h)=F^{\prime}(\alpha)\left[h+\sum_{q=2}^{p-1}C_{q}h^{q}\right]+O(h^{p}) \tag{8}\]  
where \(C_{q}=(1/q! )[F^{\prime}(\alpha)]^{-1}F^{(q)}(\alpha)\), \(q\geq 2\). It is noted that \(C_{q}h^{q}\in\mathbb{R}^{n}\) since \(F^{(q)}(\alpha)\in\mathcal{L}(\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}, \mathbb{R}^{n})\) and \([F^{\prime}(\alpha)]^{-1}\in\mathcal{L}(\mathbb{R}^{n})\). Also, we can expand \(F^{\prime}(\alpha+h)\) in Taylor series  
\[F^{\prime}(\alpha+h)=F^{\prime}(\alpha)\left[I+\sum_{q=2}^{p-1}qC_{q}h^{q-1} \right]+O(h^{p}) \tag{9}\]  
where \(I\) is the identity matrix. It is also noted that \(qC_{q}h^{q-1}\in\mathcal{L}(\mathbb{R}^{n})\). Denote \(e^{(k)}=x^{(k)}-\alpha\), so the error at the \((k+1)\)th iteration is \(e^{(k+1)}={L{e^{(k)}}^{p}}+O({e^{(k)}}^{p+1})\), where \(L\) is a \(p\)-linear function \(L\in\mathcal{L}(\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n},\mathbb{R}^{n})\) is called the _error equation_ and \(p\) is the _order of convergence_. Observe that \({e^{(k)}}^{p}\) is \((e^{(k)},e^{(k)},\cdots,e^{(k)})\). In order to prove the convergence order for the Equation (6), we need to recall some important definitions and results from the theory of point of attraction. **Definition** (**Point of Attraction**). : _[_11_]_ _Let \(G:D\subset\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\). Then \(\alpha\) is a point of attraction of the iteration_  
\[x^{(k+1)}=G(x^{(k)}),\,k=0,1,... \tag{10}\]  
_if there is an open neighborhood \(S\) of \(\alpha\) defined by_  
\[S(\alpha)=\{x\in\mathbb{R}^{n}\Big{|}\left\|x-\alpha\right\|<\delta\},\; \delta>0,\]  
_such that \(S\subset D\) and, for any \(x^{(0)}\in S\), the iterating \(\{x^{(k)}\}\) defined by Equation (10) all lie in \(D\) and converge to \(\alpha\)._  
**Theorem 1** (**Ostrowski Theorem**). : _[_11_]_ _Assume that \(G:D\subset\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) has a fixed point \(\alpha\in int(D)\) and \(G(x)\) is Frechet differentiable on \(\alpha\). If_  
\[\rho(G^{\prime}(\alpha))=\sigma<1 \tag{11}\]  
_then \(\alpha\) is a point of attraction for \(x^{(k+1)}=G(x^{(k)})\)._  
We now prove a general result that shows \(\alpha\) is a point of attraction of a general iteration function \(G(x)=P(x)-Q(x)R(x)\). **Theorem 2**. : _Let \(F:D\subset\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\) be sufficiently Frechet differentiable at each point of an open convex neighborhood \(D\) of \(\alpha\in D\), which is a solution of the system \(F(x)=0\). Suppose that \(P,Q,R:D\subset\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) are sufficiently Frechet differentiable functionals (depending on \(F\)) at each point in \(D\) with \(P(\alpha)=\alpha\), \(Q(\alpha)\
eq 0\) and \(R(\alpha)=0\). Then, there exists a ball_  
\[S=\overline{S}(\alpha,\delta)=\left\{\left\|\alpha-x\right\|\leq\delta\right\} \subset S_{0},\;\delta>0,\]_on which the mapping_  
\[G:S\rightarrow\mathbb{R}^{n},\quad G(x)=P(x)-Q(x)R(x),\,\text{for all}x\in\ S\]  
_is well-defined; moreover, \(G\) is Frechet differentiable at \(\alpha\), thus_  
\[G^{\prime}(\alpha)=P^{\prime}(\alpha)-Q(\alpha)R^{\prime}(\alpha).\]  
**Proof:** Clearly, \(G(\alpha)=\alpha\).