## 6 Convergence Analysis  
We will only provide the proof of the convergence of iterative method Equation (23), and the convergence proofs of other iterative methods are similar. To perform the convergence analysis, we use the symbolic algebraic tools of Maple(r) 2015. **Theorem 1**. : _Let \(\mathbf{F}:\Gamma\subseteq\mathbb{R}^{\mathrm{n}}\to\mathbb{R}^{\mathrm{n}}\) be a sufficiently Frechet differentiable function on an open convex neighborhood \(\Gamma\) of \(\mathbf{u}^{*}\in\mathbb{R}^{\mathrm{n}}\) with \(\mathbf{F}(\mathbf{u}^{*})=0\) and \(\det(\mathbf{F}^{\prime}(\mathbf{u}^{*}))\
eq 0\), where \(\mathbf{F}^{\prime}(\mathbf{u})\) denotes the Frechet derivative of \(\mathbf{F}(\mathbf{u})\). Let \(\mathbf{D}_{1}=\mathbf{F}^{\prime}(\mathbf{u}^{*})\) and \(\mathbf{D}_{8}=\frac{1}{3! }\,\mathbf{F}^{\prime}(\mathbf{u}^{*})^{-1}\,\mathbf{ F}^{(\mathrm{s})}(\mathbf{u}^{*})\), for \(\mathrm{s}\geq 2\), where \(\mathbf{F}^{(\mathrm{s})}(\mathbf{u})\) denotes the s-order Frechet derivative of \(\mathbf{F}(\mathbf{u})\). Then, the iterative method Equation (23), with an initial guess in the neighborhood of \(\mathbf{u}^{*}\), converges to \(\mathbf{u}^{*}\) with local order of convergence at least six and error:_  
\[\mathbf{e}_{2}=\mathbf{Le}_{0}{}^{6}+\mathrm{O}\left(\mathbf{e}_{0}{}^{7}\right)\]  
_where \(\mathbf{e}_{0}=\mathbf{u}_{0}-\mathbf{u}^{*}\), \(\mathbf{e}_{0}{}^{\mathrm{P}}=\frac{\mathrm{P}\ times}{\left(\mathbf{e}_{0}, \mathbf{e}_{0},\ldots,\mathbf{e}_{0}\right)}\) and \(\mathbf{L}=\left(2\mathbf{D}_{2}^{2}\mathbf{D}_{3}\mathbf{D}_{2}-9\mathbf{D}_ {2}\mathbf{D}_{3}\mathbf{D}_{2}^{2}+14\mathbf{D}_{3}\mathbf{D}_{2}^{3}+16 \mathbf{D}_{2}^{6}-88\mathbf{D}_{2}^{3}+6\,\text{times}\right.\)\(\left.6\ times\right.\)\(\left.6\ times\right.\)\(\left.114\mathbf{D}_{2}^{5}\right)\) is a six-linear function, i.e., \(\mathbf{L}\in\widehat{\left(\mathbb{R}^{\mathrm{n}},\mathbb{R}^{\mathrm{n}}, \mathbb{R}^{\mathrm{n}},\cdots,\mathbb{R}^{\mathrm{n}}\right)}\) with \(\mathbf{Le}_{0}{}^{6}\in\mathbb{R}^{\mathrm{n}}\)._  
Proof. : Let \(\mathbf{F}:\Gamma\subseteq\mathbb{R}^{\mathrm{n}}\to\mathbb{R}^{\mathrm{n}}\) be a sufficiently Frechet differentiable function in \(\Gamma\). The p-th Frechet derivative of \(\mathbf{F}\) at \(\mathrm{v}\in\mathbb{R}^{\mathrm{n}}\), \(\mathrm{p}\geq 1\), is a p-linear function \(\mathbf{F}^{(\mathrm{p})}(\mathrm{v}):\widehat{\mathbb{R}^{\mathrm{n}}\mathbb{ R}^{\mathrm{n}}\cdots\mathbb{R}^{\mathrm{n}}}\) with \(\mathbf{F}^{(\mathrm{p})}(\mathrm{v})(\mathrm{h}_{1},\mathrm{h}_{2},\cdots, \mathbf{h}_{\mathrm{q}})\in\mathbb{R}^{\mathrm{n}}\). [PERSON]'s series expansion of \(\mathbf{F}(\mathbf{u}_{0})\) around \(\mathbf{u}^{*}\) is:  
\[\mathbf{F}(\mathbf{u}_{0}) =\mathbf{F}\left(\mathbf{u}^{*}+\mathbf{u}_{0}-\mathbf{u}^{*} \right)=\mathbf{F}(\mathbf{u}^{*}+\mathbf{e}_{0})\] \[=\mathbf{F}\left(\mathbf{u}^{*}\right)+\mathbf{F}^{\prime}( \mathbf{u}^{*})\,\mathbf{e}_{0}+\frac{1}{2!}\,\mathbf{F}^{\prime\prime}( \mathbf{u}^{*})\,\mathbf{e}_{0}^{2}+\frac{1}{3!}\,\mathbf{F}^{(3)}(\mathbf{u} ^{*})\,\mathbf{e}_{0}^{3}+\frac{1}{4!}\,\mathbf{F}^{(4)}(\mathbf{u}^{*})\, \mathbf{e}_{0}^{4}+\cdots\] \[=\mathbf{F}^{\prime}(\mathbf{u}^{*})\left(\mathbf{e}_{0}+\frac{1} {2!}\,\mathbf{F}^{\prime}(\mathbf{u}^{*})^{-1}\,\mathbf{F}^{\prime\prime}( \mathbf{u}^{*})\,\mathbf{e}_{0}^{2}+\frac{1}{3!}\,\mathbf{F}^{\prime}( \mathbf{u}^{*})^{-1}\,\mathbf{F}^{(3)}(\mathbf{u}^{*})\,\mathbf{e}_{0}^{3}+ \frac{1}{4!}\,\mathbf{F}^{\prime}(\mathbf{u}^{*})^{-1}\,\mathbf{F}^{(4)}( \mathbf{u}^{*})\,\mathbf{e}_{0}^{4}+\cdots\right)\] \[=\mathbf{D}_{1}\left(\mathbf{e}_{0}+\mathbf{D}_{2}\,\mathbf{e}_{0 }^{2}+\mathbf{D}_{3}\,\mathbf{e}_{0}^{3}+\mathbf{D}_{4}\,\mathbf{e}_{0}^{4}+ \mathrm{O}\left(\mathbf{e}_{0}{}^{5}\right)\,\right)\]  
where \(\mathbf{e}_{\mathrm{k}}=\mathbf{u}_{\mathrm{k}}-\mathbf{u}^{*}\).