## 5 Extension to a Family of the Sixth-Order Methods for Nonlinear Systems of Equations
### Test Example 3
**Test Example 4**  
A \(d\)-dimensional nonlinear equation \(f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}\) with \(d=max(4,n\in\mathbb{N})\) is given by:  
\[0=x_{i}-\cos(x_{i}-\sum_{j=1}^{4}x_{j}),i=1,2,\cdots,d\in\mathbb{N}. \tag{46}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(\gamma\)} & \multirow{2}{*}{**MT**} & \multicolumn{3}{c}{**1**} & \multicolumn{1}{c}{**2**} & \multicolumn{1}{c}{**3**} \\ \hline  & & \(\|x^{(k+1)}-x^{(k)}\|\|\) & 2.230120 \(\times 10^{-1}\) & 6.736388 \(\times 10^{-13}\) & 4.536504 \(\times 10^{-82}\) \\  & EM1 & \(\|f(x^{(k+1)})\|\) & 9.957324 \(\times 10^{-13}\) & 7.141648 \(\times 10^{-82}\) & 1.826912 \(\times 10^{-154}\) \\  & & ACOC & - & - & 6.004539 \\ \hline  & & \(\|x^{(k+1)}-x^{(k)}\|\) & 2.230120 \(\times 10^{-1}\) & 1.267438 \(\times 10^{-12}\) & 4.035285 \(\times 10^{-80}\) \\  & EM2 & \(\|f(x^{(k+1)})\|\) & 1.767457 \(\times 10^{-12}\) & 6.182344 \(\times 10^{-80}\) & 3.498271 \(\times 10^{-154}\) \\  & ACOC & - & - & - & 6.002191 \\ \hline  & & \(\|x^{(k+1)}-x^{(k)}\|\) & 2.230120 \(\times 10^{-1}\) & 2.766996 \(\times 10^{-12}\) & 1.050932 \(\times 10^{-77}\) \\  & EM3 & \(\|f(x^{(k+1)})\|\) & 3.768017 \(\times 10^{-12}\) & 1.581912 \(\times 10^{-77}\) & 2.473651 \(\times 10^{-154}\) \\  & ACOC & - & - & - & 5.99896 \\ \hline  & & \(\|x^{(k+1)}-x^{(k)}\|\) & 7.495256 \(\times 10^{-3}\) & 1.558196 \(\times 10^{-21}\) & 2.153197 \(\times 10^{-133}\) \\  & EM4 & \(\|f(x^{(k+1)})\|\) & 2.256546 \(\times 10^{-21}\) & 3.262086 \(\times 10^{-133}\) & 1.826912 \(\times 10^{-154}\) \\  & ACOC & - & - & - & 5.987504 \\ \hline  & & \(\|x^{(k+1)}-x^{(k)}\|\) & 2.230120 \(\times 10^{-1}\) & 3.364258 \(\times 10^{-13}\) & 4.136278 \(\times 10^{-84}\) \\
2/3 & LKI & \(\|f(x^{(k+1)})\|\) & 4.578823 \(\times 10^{-13}\) & 5.464893 \(\times 10^{-84}\) & 1.826912 \(\times 10^{-154}\) \\  & ACOC & - & - & - & 5.99847 \\ \hline  & & \(\|x^{(k+1)}-x^{(k)}\|\) & 2.230120 \(\times 10^{-1}\) & 3.625463 \(\times 10^{-12}\) & 6.626067 \(\times 10^{-77}\) \\  & LK2 & \(\|f(x^{(k+1)})\|\) & 4.879379 \(\times 10^{-12}\) & 9.868594 \(\times 10^{-77}\) & 3.498271 \(\times 10^{-154}\) \\  & ACOC & - & - & - & 6.000400 \\ \hline  & & \(\|x^{(k+1)}-x^{(k)}\|\) & 2.230120 \(\times 10^{-1}\) & 1.690241 \(\times 10^{-12}\) & 3.277540 \(\times 10^{-79}\) \\  & LK3 & \(\|f(x^{(k+1)})\|\) & 2.350565 \(\times 10^{-21}\) & 5.011579 \(\times 10^{-79}\) & 2.473651 \(\times 10^{-154}\) \\  & ACOC & - & - & - & 5.999111 \\ \hline  & & \(\|x^{(k+1)}-x^{(k)}\|\) & 2.230120 \(\times 10^{-1}\) & 4.876737 \(\times 10^{-12}\) & 5.664655 \(\times 10^{-76}\) \\  & LK4 & \(\|f(x^{(k+1)})\|\) & 6.554988 \(\times 10^{-12}\) & 8.425112 \(\times 10^{-76}\) & 4.775668 \(\times 10^{-154}\) \\  & ACOC & - & - & - & 5.997538 \\ \hline  & & \(\|x^{(k+1)}-x^{(k)}\|\) & 2.230120 \(\times 10^{-1}\) & 1.633145 \(\times 10^{-11}\) & 2.761720 \(\times 10^{-72}\) \\  & LK5 & \(\|f(x^{(k+1)})\|\) & 2.189664 \(\times 10^{-11}\) & 4.097352 \(\times 10^{-72}\) & 2.790655 \(\times 10^{-154}\) \\  & ACOC & - & - & - & 5.996055 \\ \hline  & & \(\|x^{(k+1)}-x^{(k)}\|\) & 2.230120 \(\times 10^{-1}\) & 1.263880 \(\times 10^{-13}\) & 2.738954 \(\times 10^{-87}\) \\  & EM5 & \(\|f(x^{(k+1)})\|\) & 3.328247 \(\times 10^{-13}\) & 5.839804 \(\times 10^{-87}\) & 2.473651 \(\times 10^{-154}\) \\  & ACOC & - & - & - & 6.015056 \\ \hline  & & \(\|x^{(k+1)}-x^{(k)}\|\) & 2.230120 \(\times 10^{-1}\) & 1.440429 \(\times 10^{-12}\) & 3.815658 \(\times 10^{-75}\) \\  & EM6 & \(\|f(x^{(k+1)})\|\) & 4.931396 \(\times 10^{-12}\) & 5.169287 \(\times 10^{-75}\) & 2.790655 \(\times 10^{-154}\) \\  & ACOC & - & - & - & 5.592300 \\ \hline  & & \(\|x^{(k+1)}-x^{(k)}\|\) & 2.230120 \(\times 10^{-1}\) & 9.195040 \(\times 10^{-14}\) & 9.072776 \(\times 10^{-88}\) \\  & EM7 & \(\|f(x^{(k+1)})\|\) & 3.009661 \(\times 10^{-13}\) & 1.315517 \(\times 10^{-87}\) & 2.473651 \(\times 10^{-154}\) \\  & ACOC & - & - & - & 5.975547 \\ \hline
1 & LK6 & \(\|x^{(k+1)}-x^{(k)}\|\) & 2.230120 \(\times 10^{-1}\) & 1.540472 \(\times 10^{-12}\) & 4.642753 \(\times 10^{-74}\) \\  & ACOC & - & - & - & 2.790655 \(\times 10^{-154}\) \\ \hline  & & \(\|x^{(k+1)}-x^{(k)}\|\) & 5.029171 \(\times 10^{-12}\) & 6.674639 \(\times 1 The above nonlinear system is described in [4].