## 4 Numerical Testing  
The two preconditioners \(\omega(z)\) and \(\lambda(z)\) produce families of iterative methods. If we define \(\omega(z)=exp(\varpi\,z)\) and \(\lambda(z)=exp(\theta z)\), we get the following two-parameter family of iterative methods for solving nonlinear equations that have zeros with unknown multiplicity. \[\text{S1:}\ z_{k+1}=z_{k}-\frac{\left(\varpi\,\phi(z)+\phi^{\prime}(z)\right) \phi(z)}{\left(\varpi-\varpi\right)\phi^{\prime}(z)\left(\varpi+\phi(z) \right)\phi^{\prime}(z)^{2}-\phi^{\prime\prime}(z)\,\phi(z)}.\]  
Now we choose \(\omega(z)=exp(\varpi\,\phi(z))\) and \(\lambda(z)=\left(\vartheta\,\phi(z)\right)\), and obtain the following method  
\[\text{S2:}\ z_{k+1}=z_{k}-\frac{\phi^{\prime}(z)\left(1+\varpi\,\phi(z) \right)\phi(z)}{\phi^{\prime}(z)^{2}\left(1+\left(\vartheta-\varpi\right) \left(1+\varpi\,\phi(z)\right)\phi(z)\right)-\phi^{\prime\prime}(z)\,\phi(z) \left(1+\varpi\,\phi(z)\right)}.\]  
We only conducted numerical testing for the system of nonlinear equations, and the cases for the nonlinear equations are similar. It is important to test the computational convergence order (CCO) of the proposed iterative methods. In all our simulations, we adopted the following definition of CCO:  
\[\text{CCO}=\frac{log\big{(}||\mathbf{\Phi}(\mathbf{z}_{k+1})||_{\infty}/|| \mathbf{\Phi}(\mathbf{z}_{k})||_{\infty}\big{)}}{log\big{(}||\mathbf{\Phi}( \mathbf{z}_{k})||_{\infty}/||\mathbf{\Phi}(\mathbf{z}_{k-1})||_{\infty}\big{)} }\ \text{or}\ \frac{log\big{(}||\mathbf{z}_{k+1}-\boldsymbol{\kappa}||_{\infty}/|| \mathbf{z}_{k}-\boldsymbol{\kappa}||_{\infty}\big{)}}{log\big{(}||\mathbf{z}_{ k}-\boldsymbol{\kappa}||_{\infty}/||\mathbf{z}_{k-1}-\boldsymbol{\kappa}||_{ \infty}\big{)}}. \tag{32}\]  
For numerical simulations, three problems were selected with different multiplicities. The performance of iterative method (11) is not better, comparatively. The various choices for the preconditioners are made in Tables 1-3 for all three problems. In Table 1, we have shown that the selection of preconditioners has an influence on the numerical accuracy of computed zeros with multiplicities. Moreover, the computational cost of performing the different operation is reasonable, because in all cases, we selected preconditioners in a way that their first- and second-order Frechet derivatives are diagonal matrices. When we selected \(\mathbf{\Lambda}(\mathbf{z})=6+cos(\mathbf{z})/10\) and \(\mathbf{\Omega}(\mathbf{z})=6+cos(\mathbf{z})/10\) for Problem 1, we achieved the best accuracy in computed zeros with different multiplicities. For the second problem, Table 2 shows that the selection of \(\mathbf{\Lambda}(\mathbf{z})\) produces good accuracy. In Table 3, again the selection of both preconditioners provides the best accuracy, comparatively. \[\text{Problem 1}= \begin{cases}&\[PERSON]_{1}(\mathbf{z})=(z_{1}-1)^{4}\,exp(z_{2})= 0\\ &\[PERSON]_{2}(\mathbf{z})=(z_{2}-2)^{5}\,(z_{1}\,z_{2}-1)=0\\ &\[PERSON]_{3}(\mathbf{z})=(z_{3}+4)^{6}=0\end{cases}  \tag{33}\]  
\[\text{Problem 2}= \begin{cases}\[PERSON]_{1}(\mathbf{z})=z_{1}\,z_{2}=0\\ \[PERSON]_{2}(\mathbf{z})=z_{2}\,z_{3}=0\\ \[PERSON]_{3}(\mathbf{z})=z_{3}\,z_{4}=0\\ \Phi_{4}(\mathbf{z})=z_{4}\,z_{1}=0\end{cases}  \tag{34}\]  
\[\text{Problem 3}= \begin{cases}\Phi_{1}(\mathbf{z})=\sqrt{z_{1}-1}\,z_{2}\,z_{3}= 0\\ \[PERSON]_{2}(\mathbf{z})=\sqrt{z_{2}-1}\,z_{1}\,z_{3}=0\\ \[PERSON]_{3}(\mathbf{z})=\sqrt{z_{3}-1}\,z_{1}\,z_{2}=0\end{cases}  \tag{35}\]