## 3 Construction of Iterative Methods
### Iterative Method III  
The summation of \(\mathbf{x}_{0}\), \(\mathbf{x}_{1}\), \(\mathbf{x}_{2}\) and \(\mathbf{x}_{3}\) induces the following iterative method:  
\[\left\{ \begin{array}{l}\mathbf{u}_{0}=\text{Initial guess}\\ \mathbf{F}^{\prime}(\mathbf{u}_{0})\,\boldsymbol{\varphi}_{1}=\mathbf{F}( \mathbf{u}_{0})\\ \mathbf{u}_{1}=\mathbf{u}_{0}-\boldsymbol{\varphi}_{1}\\ \mathbf{F}^{\prime}(\mathbf{u}_{0})\,\boldsymbol{\varphi}_{2}=\mathbf{F}( \mathbf{u}_{1})\\ \mathbf{F}^{\prime}(\mathbf{u}_{0})\,\boldsymbol{\varphi}_{3}=\mathbf{F}^{ \prime}(\mathbf{u}_{1})\,\boldsymbol{\varphi}_{2}\\ \mathbf{F}^{\prime}(\mathbf{u}_{0})\,\boldsymbol{\varphi}_{4}=\mathbf{F}^{ \prime}(\mathbf{u}_{1})\,\boldsymbol{\varphi}_{3}\\ \mathbf{F}^{\prime}(\mathbf{u}_{0})\,\boldsymbol{\varphi}_{5}=\mathbf{F}^{ \prime\prime}(\mathbf{u}_{1})\,\boldsymbol{\varphi}_{2}^{2}\\ \mathbf{u}_{2}=\mathbf{u}_{1}+(-2+\alpha_{0})\,\boldsymbol{\varphi}_{2}+(1-2 \,\alpha_{0})\,\boldsymbol{\varphi}_{3}+\alpha_{0}\,\boldsymbol{\varphi}_{4}+( -2\alpha_{0}-5/2)\,\boldsymbol{\varphi}_{5}\end{array} \right. \tag{22}\]  
The order of convergence of the iterative method Equation (22) is five. We extend the model of iterative method Equation (22), by including more free-parameters, so that we can enhance the convergence order. \[\left\{ \begin{array}{l}\mathbf{u}_{0}=\text{Initial guess}\\ \mathbf{F}^{\prime}(\mathbf{u}_{0})\,\boldsymbol{\varphi}_{1}=\mathbf{F}( \mathbf{u}_{0})\\ \mathbf{u}_{1}=\mathbf{u}_{0}-\boldsymbol{\varphi}_{1}\\ \mathbf{F}^{\prime}(\mathbf{u}_{0})\,\boldsymbol{\varphi}_{2}=\mathbf{F}( \mathbf{u}_{1})\\ \mathbf{F}^{\prime}(\mathbf{u}_{0})\,\boldsymbol{\varphi}_{3}=\mathbf{F}^{ \prime}(\mathbf{u}_{1})\,\boldsymbol{\varphi}_{2}\\ \mathbf{F}^{\prime}(\mathbf{u}_{0})\,\boldsymbol{\varphi}_{4}=\mathbf{F}^{ \prime}(\mathbf{u}_{1})\,\boldsymbol{\varphi}_{3}\\ \mathbf{F}^{\prime}(\mathbf{u}_{0})\,\boldsymbol{\varphi}_{5}=\mathbf{F}^{ \prime\prime}(\mathbf{u}_{1})\,\boldsymbol{\varphi}_{2}^{2}\\ \mathbf{F}^{\prime}(\mathbf{u}_{0})\,\boldsymbol{\varphi}_{6}=\mathbf{F}^{ \prime\prime}(\mathbf{u}_{1})\,\boldsymbol{\varphi}_{2}\,\boldsymbol{\varphi}_{3 }\\ \mathbf{u}_{2}=\mathbf{u}_{1}+\beta_{1}\,\boldsymbol{\varphi}_{2}+\beta_{2}\, \boldsymbol{\varphi}_{3}+\beta_{3}\,\boldsymbol{\varphi}_{4}+\beta_{4}\, \boldsymbol{\varphi}_{5}+\beta_{5}\,\boldsymbol{\varphi}_{6}\end{array} \right. \tag{23}\]  
If \(\beta_{1}=-3\), \(\beta_{2}=3\), \(\beta_{3}=-1\), \(\beta_{4}=-4\) and \(\beta_{5}=\frac{7}{2}\), then the order of convergence of the iterative method Equation (23) is six. Further iterative methods can be constructed by adding more \(\mathbf{x}_{i}^{\prime}\)s and following essentially the same basic ideas. Application to Systems of Nonlinear Equations Associated with Ordinary and Partial Differential Equations  
According to our analysis, we observed that the parametric homotopy method has the potential to provide an effective model for the construction of a higher order iterative method for solvingsystems of nonlinear equations. The unknown parameters in the homotopy method and the extended models (bases on homotopy provided models) can be used to enhance the convergence order of the iterative method. The iterative method Equation (21) can be employed effectively for solving a general system of nonlinear equations. On the contrary, the iterative method Equation (23) is only efficient to solve weakly nonlinear systems associated with nonlinear ordinary equations (ODEs) and partial differential equations (PDEs). In fact, many ODEs and PDEs can be written as:  
\[\mathrm{L}(\mathrm{y})+\mathrm{f}(\mathrm{y})-\mathrm{w}(\mathrm{x})=0 \tag{24}\]  
where \(\mathrm{L}(\cdot)\) is a linear differential operator, \(\mathrm{f}(\cdot)\) is a nonlinear function, \(\mathrm{y}\) is a dependent variable and \(\mathrm{x}\) is an independent variable.