## 3 Special Cases of Weight Functions
### Global Convergence
We then take \(601\times 601\) equally spaced points in the square as  
\begin{table}
\begin{tabular}{c c|c c c c c c c} \hline \multirow{2}{*}{\(\gamma\)} & \multirow{2}{*}{**Method**} & \multirow{2}{*}{\(|x_{n}-\alpha|\)} & \multicolumn{6}{c}{\(f(x);x_{0}\)} \\  & & & \(f_{1}\)**0.01** & \(f_{2}\)**2.0** & \(f_{3}\)**1.53** & \(f_{4}\)**0.73** & \(f_{5}\)**1.87** & \(f_{6}\)**0.52 + 0.85i** & \(f_{72}\)**1.05** \\ \hline \multirow{8}{*}{\(2/3\)} & **EM1** & \(|x_{1}-\alpha|\) & 1.35-12 \({}^{*}\) & 4.03e-13 & 5.07e-9 & 1.64e-12 & 3.13e-5 & 9.14e-10 & 2.26e-9 \\  & & \(|x_{2}-\alpha|\) & 7.50e-72 & 2.30e-77 & **1.99e-50** & **2.49e-71** & 2.59e-26 & 2.72e-54 & 2.34e-53 \\ \cline{2-10}  & **EM2** & \(|x_{1}-\alpha|\) & 2.54e-12 & 7.48e-13 & 1.11e-8 & 4.50e-12 & 3.92e-5 & 8.81e-10 & 3.89e-9 \\  & & \(|x_{2}-\alpha|\) & 6.61e-70 & 1.75e-75 & 5.43e-48 & 2.97e-68 & 1.63e-25 & 2.16e-54 & 1.11e-51 \\ \cline{2-10}  & **EM3** & \(|x_{1}-\alpha|\) & 5.88e-12 & 1.68e-12 & 3.05e-8 & 1.49e-11 & 5.62e-5 & 8.08e-10 & 8.09e-9 \\  & & \(|x_{2}-\alpha|\) & 2.26e-67 & 5.13e-73 & 6.77e-45 & 1.34e-64 & 2.73e-24 & **1.28e-54** & 1.94e-49 \\ \cline{2-10}  & **EM4** & \(|x_{1}-\alpha|\) & 4.17e-12 & 1.20e-12 & 1.89e-8 & 8.28e-12 & 4.89e-5 & 8.49e-10 & 6.03e-9 \\  & & \(|x_{2}-\alpha|\) & 2.05e-68 & 4.97e-74 & 2.37e-46 & 2.14e-66 & 9.32e-25 & 1.76e-54 & 2.45e-50 \\ \cline{2-10}  & **LK1** & \(|x_{1}-\alpha|\) & 6.33e-13 & 1.78e-13 & 6.13e-9 & 3.26e-12 & 1.37e-5 & 9.89e-10 & 6.46e-10 \\  & & \(|x_{2}-\alpha|\) & **3.58e-74** & **8.08e-80** & 8.66e-50 & 3.13e-69 & **9.64e-30** & 4.55e-54 & **4.72e-57** \\ \hline \multirow{2}{*}{\(2/3\)} & **LK2** & \(|x_{1}-\alpha|\) & 7.48e-12 & 2.10e-12 & 3.23e-8 & 1.561e-11 & 6.43e-5 & 8.34e-10 & 1.00e-8 \\  & & \(|x_{2}-\alpha|\) & 1.20e-66 & 2.51e-72 & 1.29e-44 & 1.86e-64 & 8.09e-24 & 1.78e-54 & 9.18e-49 \\ \cline{2-10}  & **LK3** & \(|x_{1}-\alpha|\) & 3.59e-12 & 1.04e-12 & 1.79e-8 & 8.13e-12 & 4.50e-5 & 8.39e-10 & 5.22e-9 \\  & & \(|x_{2}-\alpha|\) & 7.27e-69 & 1.80e-74 & 1.55e-46 & 1.87e-66 & 4.76e-25 & 1.57e-54 & 8.82e-51 \\ \hline \multirow{2}{*}{\(2/3\)} & **LK4** & \(|x_{1}-\alpha|\) & 1.05e-11 & 2.93e-12 & 5.35e-8 & 2.82e-11 & 7.39e-5 & 7.94e-10 & 1.34e-8 \\  & & \(|x_{2}-\alpha|\) & 1.32e-65 & 2.59e-71 & 3.71e-43 & 1.17e-62 & 2.37e-23 & 1.29e-54 & 7.28e-48 \\ \cline{2-10}  & **LK5** & \(|x_{1}-\alpha|\) & 3.58e-11 & 9.46e-12 & 1.94e-7 & 1.24e-10 & 1.27e-4 & 6.41e-10 & 3.85e-8 \\  & & \(|x_{2}-\alpha|\) & 6.72e-62 & 9.48e-68 & 3.57e-39 & 4.05e-58 & 1.74e-21 & 3.79e-55 & 1.24e-44 \\ \hline \multirow{2}{*}{\(1\)} & **EM5** & \(|x_{1}-\alpha|\) & 2.02e-12 & 3.88e-13 & 2.72e-8 & 2.23e-11 & 2.60e-5 & 2.17e-9 & 1.88e-9 \\  & & \(|x_{2}-\alpha|\) & 1.16e-70 & 1.99e-77 & 2.91e-45 & 2.25e-63 & 2.11e-26 & 1.19e-51 & 1.16e-53 \\ \cline{2-10}  & **EM6** & \(|x_{1}-\alpha|\) & 1.38e-12 & 3.93e-13 & 2.88e-9 & 8.25e-13 & 1.33e-5 & 2.32e-9 & 1.96e-9 \\  & & \(|x_{2}-\alpha|\) & 9.18e-72 & 1.94e-77 & 3.98e-52 & 2.26e-73 & 1.06e-88 & 1.49e-51 & 9.28e-54 \\ \cline{2-10}  & **EM7** & \(|x_{1}-\alpha|\) & 4.19e-13 & 8.51e-14 & 5.45e-9 & 3.56e-12 & 1.17e-5 & 2.03e-9 & 4.68e-10 \\  & & \(|x_{2}-\alpha|\) & 2.00e-75 & 4.73e-82 & 3.20e-50 & 5.72e-69 & 4.62e-9 & 6.12e-52 & 6.03e-58 \\ \cline{2-10}  & **LK6** & \(|x_{1}-\alpha|\) & 3.93e-12 & 1.12e-12 & 1.81e-8 & 7.70e-12 & 5.75e-5 & 1.91e-9 & 5.60e-9 \\  & & \(|x_{2}-\alpha|\) & 1.36e-68 & 3.03e-74 & 1.65e-46 & 1.27e-66 & 2.27e-24 & **4.21e-52** & 1.41e-50 \\ \cline{2-10}  & **LK7** & \(|x_{initial points for methods (1). We color the point based on the root it converged to. This way we can figure out if the method converged within the maximum number of iteration and if it converged to the root closer to the initial point. Figures 1-4 present basins of attraction of 17 iterative maps when applied to various polynomials. **Example 1. **_As a first example, we have taken a quadratic polynomial with all real roots:_  
\[p_{1}(z)=(z^{2}-1). \tag{26}\]  
Clearly the roots are \(\pm 1\).