# Local Comparison between Two Ninth Convergence Order Algorithms for Equations
## 3 Numerical Examples  
**Example 1**.: _Let us consider a system of differential equations governing the motion of an object and given by_  
\[H_{1}^{\prime}(x)=e^{x},\,H_{2}^{\prime}(y)=(e-1)y+1,\,H_{3}^{\prime}(z)=1\]  
_with initial conditions \(H_{1}(0)=H_{2}(0)=H_{3}(0)=0.\) Let \(H=(H_{1},H_{2},H_{3}).\) Let \(B_{1}=B_{2}=\mathbb{R}^{3},D=\tilde{U}(0,1),x_{*}=(0,0,0)^{T}.\) Define function \(H\) on \(D\) for \(w=(x,y,z)^{T}\) by_  
\[H(w)=(e^{x}-1,\frac{e-1}{2}y^{2}+y,z)^{T}.\]  
_The Frechet-derivative is defined by_  
\[H^{\prime}(v)=\left[\begin{array}{ccc}e^{x}&0&0\\ 0&(e-1)y+1&0\\ 0&0&1\end{array}\right].\]  
_Notice that using the (\(\mathcal{A}\)) conditions, we get for \(x_{*}=(0,0,0)^{T},\omega_{0}(s)=(e-1)s,\omega(s)=e^{\frac{1}{e-1}}s,\omega_{1 }(s)=e^{\frac{1}{e-1}}.\) The radii are_  
\[R_{1}=0.38269191223238574472986783803208,\,R_{2}=0.1924942435777614313519023880 8718,\]  
\[R_{3}=0.16097144932100204695046841152362,\,R_{4}=0.1731041505859549911594541526938,\]  
\[\bar{R}_{2}=0.2304376760127628265273584654925,\,\bar{R}_{3}=2.5823927758875332182 846750766411,\]\[\bar{R}_{4}=0.2195161774302133161906880332026.\]  
**Example 2**.: _Let \(B_{1}=B_{2}=C[0,1]\), the space of continuous functions defined on \([0,1]\) be equipped with the max norm. Let \(D=\overline{U}(0,1)\). Define function \(H\) on \(D\) by_  
\[H(\varphi)(x)=\varphi(x)-5\int_{0}^{1}x\theta\varphi(\theta)^{3}d\theta. \tag{27}\]  
_We have that_  
\[H^{\prime}(\varphi(\xi))(x)=\xi(x)-15\int_{0}^{1}x\theta\varphi(\theta)^{2} \xi(\theta)d\theta,\ \ \text{for each}\ \ \xi\in D.\]  
_Then, we get \(x_{*}=0\), so \(\omega_{0}(s)=7.5s,\omega(s)=15s\) and \(\omega_{1}(s)=2\). Then the radii are_  
\[R_{1}=0.066666666666666666666666666667,R_{2}=0.035324865589989970504625205194316,\]  
\[R_{3}=0.047263681322789477534662694324652,R_{4}=0.021857976760806939464654163884916,\]  
\[\bar{R}_{2}=1.1302558424873363485119170945836,\bar{R}_{3}=0.13819337319040553291316 086870211,\]  
\[\bar{R}_{4}=0.052052957742070200819473058118092.\]  
**Example 3**.: _Returning back to the motivational example at the introduction of this study, we have for \(x_{*}=1\), \(\omega_{0}(s)=\omega(s)=96.6629073s\) and \(\omega_{1}(s)=2\). Then, the radii are_  
\[R_{1}=0.0068968199414654552878434223828208,R_{2}=0.00077090035103644658290300561986 896,\]  
\[R_{3}=0.00012680765288154951706510453757204,R_{4}=0.0102448072794521886919039133 09034,\]  
\[\bar{R}_{2}=2.0195452298754390518809032073477,\bar{R}_{3}=0.04441223697238345924 3651770975703,\]  
\[\bar{R}_{4}=1.9996509448227068883596757586929.\]  
**Author Contributions:** Conceptualization, S.R. and I.K.A.; methodology, S.R., I.K.A. and S.G.; software, I.K.A. and S.G.; validation, S.R., I.K.A. and S.G.; formal analysis, S.R., I.K.A. and S.G.; investigation, S.R., I.K.A. and S.G.; resources, S.R., I.K.A. and S.G.; data curation, S.R. and S.G.; writing--original draft preparation, S.R., I.K.A. and S.G.; writing--review and editing, I.K.A. and S.G.; visualization, S.R., I.K.A. and S.G.; supervision, I.K.A. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Conflicts of Interest:** The authors declare no conflict of interest.