# An Optimal Order Method for Multiple Roots in Case of Unknown Multiplicity
## 3 Numerical Testing with the Conclusion  
In this portion, we employ the proposed three-step method Equation (15) (MM8) for solving five nonlinear equations and scrutinize them by the methods given by [PERSON] and [PERSON] Equation (9) (MM6), in [24], and the [PERSON] _et al._ Equation (8) (MM5), in [23]. Displayed in Tables 1-3 are the absolute error in the root, the absolute value of the function and the absolute error in the approximation of unknown multiplicity, for three iterations. All of the computations were done by using Mathematica 8. We mention below five test equations along with their exact roots \(\xi\) (the functions, as well as initial guesses are taken from [23]):  
\[1.\,f_{1} =\frac{(x-5^{1/2})^{4}}{(x-1)^{2}+1},\,\,m=4,\,\,\xi=2.2360...,\] \[2.\,f_{2} =(8 xe^{-x^{2}}-2x-3)^{8},\,\,m=8,\,\,\xi=1.7903...,\] \[3.\,f_{3} =(ln(x^{2}+3x+5)-2x+7)^{8},\,\,m=8,\,\,\xi=5.4690...,\] \[4.\,f_{4} =\frac{(x-2)^{4}}{(x-1)^{2}+1},\,\,m=4,\,\,\xi=2.000...,\] \[5.\,f_{5} =\left(x^{1/2}-\frac{1}{x}-1\right)^{7},\,\,m=7,\,\,\xi=2.1478...\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(f\) & \(n\) & \(|x_{n}-\xi|\) & \(|f(x_{n})|\) & \(|m-m_{n}|\) \\ \hline \multirow{3}{*}{\(f_{1}\)} & 1 & 7.6409e\(-\)3 & 1.9138e\(-\)3 & 1.1487e+0 \\  & 2 & 1.7779e\(-\)15 & 4.4447e\(-\)16 & 7.4627e\(-\)3 \\  & 3 & 2.8524e\(-\)92 & 7.1310e\(-\)92 & 1.7387e\(-\)15 \\ \hline \multirow{3}{*}{\(f_{2}\)} & 1 & 4.8261e\(-\)3 & 6.0480e\(-\)4 & 3.5708e\(-\)1 \\  & 2 & 3.1700e\(-\)15 & 3.9625e\(-\)16 & 2.0356e\(-\)2 \\  & 3 & 2.5528e\(-\)88 & 3.1910e\(-\)89 & 1.3377e\(-\)14 \\ \hline \multirow{3}{*}{\(f_{3}\)} & 1 & 1.2583e\(-\)7 & 1.5729e\(-\)8 & 1.4907e\(-\)1 \\  & 2 & 1.6225e\(-\)50 & 2.0282e\(-\)51 & 1.0133e\(-\)8 \\  & 3 & 7.4567e\(-\)308 & 9.3209e\(-\)309 & 1.3066e\(-\)51 \\ \hline \multirow{3}{*}{\(f_{4}\)} & 1 & 4.1460e\(-\)3 & 1.0376e\(-\)3 & 1.0366e\(+\)0 \\  & 2 & 3.9308e\(-\)17 & 9.8270e\(-\)18 & 4.1460e\(-\)3 \\  & 3 & 2.8495e\(-\)101 & 7.1237e\(-\)102 & 3.9308e\(-\)17 \\ \hline \multirow{3}{*}{\(f_{5}\)} & 1 & 4.4860e\(-\)3 & 6.4158e\(-\)4 & 1.7819e\(+\)0 \\  & 2 & 9.8136e\(-\)17 & 1.4019e\(-\)17 & 7.8955e\(-\)3 \\ \cline{1-1}  & 3 & 1.0975e\(-\)98 & 1.5679e\(-\)99 & 1.7315e\(-\)16 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical results for \(MM6\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(f\) & \(n\) & \(|x_{n}-\xi|\) & \(|f(x_{n})|\) & \(|m-m_{n}|\) \\ \hline \multirow{3}{*}{\(f_{1}\)} & 1 & 7.1979e\(-\)3 & 1.8026e\(-\)3 & 1.1485e+0 \\  & 2 & 5.9419e\(-\)14 & 1.4855e\(-\)14 & 70306e\(-\)3 \\  & 3 & 2.3260e\(-\)69 & 5.8149e\(-\)70 & 5.8109e\(-\)14 \\ \hline \multirow{3}{*}{\(f_{2}\)} & 1 & 4.7520e\(-\)3 & 5.9550e\(-\)4 & 3.5708e\(-\)1 \\  & 2 & 6.2369e\(-\)14 & 7.7961e\(-\)15 & 2.0043e\(-\)2 \\  & 3 & 2.4558e\(-\)68 & 3.0697e\(-\)69 & 2.6318e\(-\)13 \\ \hline \multirow{3}{*}{\(f_{3}\)} & 1 & 2.2101e\(-\)7 & 2.7627e\(-\)8 & 1.4907e\(-\)1 \\  & 2 & 1.8202e\(-\)42 & 2.2753e\(-\)43 & 1.7798e\(-\)8 \\  & 3 & 6.8964e\(-\)218 & 8.6205e\(-\)219 & 1.4658e\(-\)42 \\ \hline \multirow{3}{*}{\(f_{4}\)} & 1 & 3.5022e\(-\)3 & 8.7632e\(-\)4 & 1.0363e\(+\)0 \\  & 2 & 1.8093e\(-\)15 & 4.5231e\(-\)16 & 3.5022e\(-\)3 \\ \cline{1-1}  & 3 & 6.6558e\(-\)77 & 1.6639e\(-\)77 & 1.8093e\(-\)15 \\ \hline \multirow{3}{*}{\(f_{5}\)} & 1 & 4.7515e\(-\)3 & 6.7960e\(-\)4 & 1.7821e\(+\)0 \\ \cline{1-1}  & 2 & 3.6795e\(-\)15 & 5.2564e\(-\)16 & 8.3616e\(-\)3 \\ \cline{1-1}  & 3 & 1.0884e\(-\)75 & 1.5548e\(-\)76 & 6.4920e\(-\)15 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numerical results for \(MM5\). The numerical values in Tables 1-3 validate that the presented scheme MM8 performs better, not only for the absolute error in the root and the absolute value of the function as compared to MM6 and MM5, but also in the approximation of unknown multiplicity. The techniques explained in this note could also be adapted to new high order methods designed recently (see [30] and the references therein): we will explore this matter in future research.