## 3 Numerical Examples  
**Example 1**.: _Suppose that the motion of an object in three dimensions is governed by system of differential equations_  
\[f_{1}^{\prime}(x)-f_{1}(x)-1=0,\] \[f_{2}^{\prime}(y)-(e-1)y-1=0, \tag{38}\] \[f_{3}^{\prime}(z)-1=0.\]  
_with \(x\), \(y\), \(z\in D\) for \(f_{1}(0)=f_{2}(0)=f_{3}(0)=0\). Then, the solution of the system is given for \(v=(x,y,z)^{T}\) by function \(F:=(f_{1},f_{2},f_{3}):D\to\mathbb{R}^{3}\) defined by_\[F(v)=\Big{(}e^{x}-1,\frac{e-1}{2}y^{2}+y,z\Big{)}^{T}. \tag{39}\]  
_Then, the Frechet-derivative is given by_  
\[F^{\prime}(v)=\ \ \ \begin{bmatrix}e^{x}&0&0\\ 0&(e-1)y+1&0\\ 0&0&1\end{bmatrix}. \tag{40}\]  
_Then, we have that \(x_{*}=(0,0,0)^{T}\), \(w(t)=w_{1}(t)+w_{2}(t)\), \(v(t)=v_{1}(t)+v_{2}(t)\), \(w_{1}(t)=L_{1}t\), \(w_{2}(t)=L_{2}t\), \(v_{1}(t)=K_{1}t\), \(v_{2}(t)=K_{2}t\), where \(L_{1}=e-1\), \(L_{2}=e\), \(K_{1}=e-2\), \(K_{2}=e\), \(\eta=0.001\), \(\gamma=1\) and \(\mu=0.01\)._  
_After solving the equation \(h_{q}(t)=0\), we obtain the root \(r_{q}=0.124067\). Similarly, the roots of Equation (11) are: \(0.0452196\) and \(0.0933513\). So,_  
\[r=min\{0.0452196,0.0933513\}=0.0452196.\]  
_Therefore,_  
\[r=0.0452196<r_{q}=0.124067.\]  
_In addition, we have that_  
\[r^{*}=0.0452196\]  
_and (see [7])_  
\[r_{1}^{+}=0.020274.\]  
_So,_  
\[r=min\{r_{1}^{+},r^{*}\}=min\{0.020274,0.0452196\}=0.020274.\]  
_It follows that sequence \(\{x_{k}\}\) is complete, \(\{t_{k}\}\to r^{*}\) in \(D\) and as such it converges to \(x_{*}\in U(x_{0},r)=U(0,0.020274)\)._  
**Example 2**.: _Consider the system of nonlinear equation \(F(X)=0\), wherein \(F=(F_{1},\cdots,F_{n})^{T}\) and \(X=(x_{1},x_{2},\ldots,x_{n})^{T}\), with_  
\[F_{i}(X)=(3-2x_{i})x_{i}^{3/2}-x_{i-1}-2x_{i+1}+1,\ \ i=1,2,\ldots,n,\]  
_where \(x_{0}=x_{n+1}=0\) by convention. This system has a complex solution. Therefore, we consider the complex initial guess \(X_{0}=(-i,-i,\ldots,-i)\). The derivative \(F^{\prime}(X)\) is given by_  
\[F^{\prime}(X)=\begin{bmatrix}\frac{3}{2}(3-2x_{1})\sqrt{x_{1}}-2x_{1}^{3/2}& -2&\cdots&0&0\\ -1&\frac{3}{2}(3-2x_{2})\sqrt{x_{2}}-2x_{2}^{3/2}&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&-1&\frac{3}{2}(3-2x_{n})\sqrt{x_{n}}-2x_{n}^{3/2}\end{bmatrix}.\]  
It is clear that \(F^{\prime}(X)\) is sparse and positive definite. Now, we solve this nonlinear problem by the Newton-HSS method (N-HSS), (see [10]), modified Newton-HSS method (MN-HSS), (see [22]), three-step modified Newton-HSS (3 MN-HSS) and four-step modified Newton-HSS (4 MN-HSS) method. The methods are compared in error estimates, CPU time (CPU-time) and the number of iterations. We use experimentally optimal parameter values of \(\alpha\) for the methods corresponding to the problem dimension \(n=100\), \(200\), \(500\), \(1000\), see Table 1. The numerical results are displayed in Table 2. From numerical results, we observe that MN-HSS outperforms N-HSS in the sense of CPU time and the number of iterations. Note that, in this example, the results in [19] can not be applied since the operators involved are not Lipschitz. However, our results can be applied by choosing \"\(w\)\" and \"\(v\)\" functions appropriately as in Example 3.1. We leave these details to the interested readers.  
**Acknowledgments:** We would like to express our gratitude to the anonymous reviewers for their help with the publication of this paper.  
**Author Contributions:** The contribution of all the authors has been equal. All of them worked together to develop the present manuscript.  
**Conflicts of Interest:** The authors declare no conflict of interest.