## 3 Convergence Analysis
\begin{split}&[F^{\prime}(y(x^{(k)}))]^{-1}\\ &=\Bigg{[}I-N_{1}e^{(k)}+(N_{1}^{2}-N_{2}){e^{(k)}}^{2}+\Big{(}N_{1 }N_{2}+N_{2}N_{1}-N_{1}^{3}-N_{3}\Big{)}{e^{(k)}}^{3}\Bigg{]}[F^{\prime}( \alpha)]^{-1}+O({e^{(k)}}^{4})\\ &=\Big{[}I+Y_{1}e^{(k)}+Y_{2}{e^{(k)}}^{2}+Y_{3}{e^{(k)}}^{3} \Big{]}\,[F^{\prime}(\alpha)]^{-1}+O({e^{(k)}}^{4}),\\ where& Y_{1}=-\frac{2}{3}C_{2},\,Y_{2}=-\frac{8}{9}C_{2}^{2}- \frac{1}{3}C_{3},\,Y_{3}=\frac{112}{27}C_{2}^{3}-\frac{22}{9}C_{2}C_{3}-\frac{ 10}{9}C_{3}C_{2}-\frac{4}{27}C_{4}\end{split}  \tag{16}\]  
On the other hand, using Equations (14) and (16), the harmonic mean can be expressed as  
\[ \begin{split} A(x^{(k)})&=\Big{[}I-\frac{4}{3}C_{2 }e^{(k)}+\left(\frac{14}{9}C_{2}^{2}-\frac{5}{3}C_{3}\right){e^{(k)}}^{2}\\ &\quad+\left(-\frac{52}{27}C_{2}^{3}+\frac{16}{9}C_{2}C_{3}+ \frac{22}{9}C_{3}C_{2}-\frac{56}{27}C_{4}\right){e^{(k)}}^{3}\Big{]}[F^{\prime }(\alpha)]^{-1}+O({e^{(k)}}^{4})\end{split}  \tag{17}\]  
Using Equations (15) and (17), we have  
\[ \begin{split}& H_{1}(x^{(k)})A(x^{(k)})\\ &=\Bigg{[}I-C_{2}\,{e^{(k)}}+(C_{2}^{2}-C_{3})\,{e^{(k)}}^{2}+ \left(-\frac{106}{27}C_{2}^{3}+\frac{17}{9}C_{2}C_{3}+\frac{10}{9}C_{3}C_{2}- \frac{10}{9}C_{4}\right){e^{(k)}}^{3}\Bigg{]}[F^{\prime}(\alpha)]^{-1}\\ &\qquad+O({e^{(k)}}^{4})\end{split}  \tag{18}\]  
Finally, by using Equations (13) and (18) in Equation (6) with some simplifications, the error equation can be expressed as:  
\[ \begin{split}& e^{(k+1)}=x^{(k)}-\alpha-H_{1}(x^{(k)})A(x^{(k)})F(x ^{(k)})\\ &=\left(\frac{79}{27}C_{2}^{3}-\frac{8}{9}C_{2}C_{3}-\frac{1}{9} C_{3}C_{2}+\frac{1}{9}C_{4}\right){e^{(k)}}^{4}+O({e^{(k)}}^{5})\end{split}  \tag{19}\]  
Thus from Equation (19), it can be concluded that the order of convergence of the \(4^{th}HM\) method is four. For the case \(r\geq 1\) we state and prove the following theorem. **Theorem 4**. : _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 (7). Furthermore the sequence converges to \(\alpha\) with order \(2r+4\), where \(r\) is a positive integer and \(r\geq 1\)._  
**Proof:** In this case,  
\[P(x)=\mu_{j-1}(x),\quad Q(x)=H_{2}(x)A(x),\quad R(x)=F(\mu_{j-1}(x)),\,j=1,...,r.\]  
We can show by induction that  
\[\mu_{j-1}(\alpha)=\alpha,\quad\mu_{j-1}^{\prime}(\alpha)=0,\quad\forall\,j=1,...,r\]so that  
\[P(\alpha)=\mu_{j-1}(\alpha)=\alpha,\,H_{2}(\alpha)=I,\,Q(\alpha)=H_{2 }(\alpha)A(\alpha)=I[F^{\prime}(\alpha)]^{-1}=[F^{\prime}(\alpha)]^{-1}\
eq 0,\] \[R(\alpha)=F(\mu_{j-1}(\alpha))=F(\alpha)=0,\] \[P^{\prime}(\alpha)=\mu^{\prime}_{j-1}(\alpha)=0,\,R^{\prime}( \alpha)=F^{\prime}(\mu_{j-1}(\alpha))\mu^{\prime}_{j-1}(\alpha)=0,\] \[G^{\prime}(\alpha)=P^{\prime}(\alpha)-Q(\alpha)R^{\prime}( \alpha)=0.\]  
So \(\rho(G^{\prime}(\alpha))=0<1\) and by [PERSON]'s theorem, \(\alpha\) is a point of attraction of Equation (7).