# Hybrid Newton-like Inverse Free Algorithms for Solving Nonlinear Equations
## 6 Numerical Examples  
The examples use \(L_{n}=G^{\prime}(x_{n}),\ \Delta=I\), which are independent of \(x_{0}\) and \(x^{*}\). **Example 1**. : _The solution sought for the nonlinear system_  
\[ \begin{array}{rcl}g_{1}&=&u-0.1\sin u-0.3\cos v+0.4\\ g_{2}&=&v-0.2\cos u+0.1\sin v+0.3.\end{array} \]  
_Let \(G=(g_{1},g_{1})\). Then, the system becomes_  
\[G(s)=0\ \ for\ \ s=(\theta_{1},\theta_{2})^{T}.\]  
_Then_  
\[G^{\prime}((u,v))=\bigg{[} \begin{array}{cc}1-0.1\cos(u)&0.3\sin(v)\\ 0.2\sin(u)&0.1\cos(v)+1\end{array} \bigg{]}.\]  
_Algorithm (2)_  
\[x_{k+1}=x_{k}-G^{\prime}(x_{k})^{-1}G(x_{k}).\]  
_Algorithm (4), \(k=1,\ \Delta=I\)_  
\[ \begin{array}{c}A_{1}(x)=I+(I-G^{\prime}(x)),\\ p_{1}(x)=x-(I+(I-G^{\prime}(x)))G(x),\\ x_{n+1}=p_{1}(x_{n}).\end{array}  \tag{37}\]  
_Algorithm (4), \(k=2,\ \Delta=I\)_  
\[ \begin{array}{c}A_{2}(x)=I+(I-G^{\prime}(x))+(I-G^{\prime}(x))^{2},\\ p_{2}(x)=x-A_{2}(x)G(x),\\ x_{n+1}=p_{2}(x_{n}).\end{array}  \tag{38}\]  
_Algorithm (4), \(k=3,\ \Delta=I\)_  
\[ \begin{array}{c}A_{3}(x)=I+(I-G^{\prime}(x))+(I-G^{\prime}(x))^{2}+(I-G^{ \prime}(x))^{3},\\ p_{3}(x)=x-A_{3}(x)G(x),\\ x_{n+1}=p_{3}(x_{n}).\end{array}  \tag{39}\]  
**Algorithm (4), \(k=4\), \(\Delta=I\)**  
\[A_{4}(x)=I+(I-G^{\prime}(x))+(I-G^{\prime}(x))^{2}+(I-G^{\prime}(x ))^{3}+(I-G^{\prime}(x))^{4},\] \[p_{4}(x)=x-A_{4}(x)G(x), \tag{40}\] \[x_{n+1}=p_{4}(x_{n}).\]  
**Algorithm (4), \(k=5\), \(\Delta=I\)**  
\[A_{5}(x)=I+(I-G^{\prime}(x))+(I-G^{\prime}(x))^{2}+(I-G^{\prime}( x))^{3}+(I-G^{\prime}(x))^{4}+(I-G^{\prime}(x))^{5},\] \[p_{5}(x)=x-A_{5}(x)G(x), \tag{41}\] \[x_{n+1}=p_{5}(x_{n}).\]  
**Algorithm (4), \(k=\overline{1,5}\), \(\Delta=G^{\prime}(x_{0})\)**  
\[x_{n+1}=x_{n}-A\Delta^{-1}G(x_{n}),\] \[\Gamma=\Delta^{-1}(\Delta-G^{\prime}(x)), \tag{42}\] \[A=I+\sum_{i=1}^{k}\Gamma^{i}.\]  
Thus, the comparison shows that the behavior of method (4) is essentially the same as [PERSON]'s method (2). However, the iterates of method (4) are cheaper to obtain than [PERSON]'s. As observed in Tables 1-4, the number of iterations required for the proposed methods with k ranging from \(3\) to \(5\) closely aligns with those of [PERSON]'s method. Table 5 shows the results of calculations to determine the Computational Order of Convergence (COC) and the Approximated Computational Order of Convergence (ACOC) aiming to compare the convergence order of method (4) with the convergence order of [PERSON]'s method (2). \begin{table}
\begin{tabular}{c c c c} \hline
**Algorithm** & **Iterations** & **Algorithm** & **Iterations** \\ \hline (2) [PERSON] & 4 & (2) [PERSON] & 4 \\ (37), \(k=1\) & 6 & (42), \(k=1\) & 8 \\ (38), \(k=2\) & 5 & (42), \(k=2\) & 6 \\ (39), \(k=3\) & 4 & (42), \(k=3\) & 5 \\ (40), \(k=4\) & 4 & (42), \(k=4\) & 5 \\ (41), \(k=5\) & 4 & (42), \(k=5\) & 4 \\ \hline \end{tabular}
\end{table} 
Table 1: Number of iterations to achieve tolerance \(\varepsilon=10^{-9}\) with initial guess \(x_{0}=(1,1)\) and \(\|I-G^{\prime}(x_{0})\|=0.3129<1\). \begin{table}
\begin{tabular}{c c c c} \hline
**Algorithm** & **Iterations** & **Algorithm** & **Iterations** \\ \hline (2) Newton & 3 & (2) Newton & 3 \\ (37), \(k=1\) & 5 & (42), \(k=1\) & 3 \\ (38), \(k=2\) & 4 & (42), \(k=2\) & 3 \\ (39), \(k=3\) & 3 & (42), \(k=3\) & 3 \\ (40), \(k=4\) & 3 & (42), \(k=4\) & 3 \\ (41), \(k=5\) & 3 & (42), \(k=5\) & 3 \\ \hline \end{tabular}
\end{table} 
Table 2: Number of iterations to achieve tolerance \(\varepsilon=10^{-9}\) with initial guess \(x_{0}=(0,0)\) and \(\|I-G^{\prime}(x_{0})\|=0.1414<1\). **Definition 2**. : _The computational order of convergence of a sequence \(\{x_{n}\}_{n\geq 0}\) is defined by_  
\[\overline{\rho}_{n}=\frac{\ln|e_{n+1}/e_{n}|}{\ln|e_{n}/e_{n-1}|},\]  
_where \(x_{n-1},x_{n},x_{n+1}\) are three consecutive iterations near the root \(\alpha\) and \(e_{n}=x_{n}-\alpha\)[6]._  
**Definition 3**.