# An Efficient Sixth-Order Newton-Type Method for Solving Nonlinear Systems
## 5 Conclusions  
In this paper, we have proposed a new iterative method of order six for solving nonlinear systems. Although five linear systems are required to be solved in each iteration, the LU decomposition of the linear systems are computed only once per full iteration. Numerical results are given to show that our method has a similar convergence behavior as the existing methods in this paper. The new method is suitable for solving small- and medium-sized systems. In order to obtain an efficient iterative method for solving nonlinear systems, we should make the iterative method achieve an as high as possible convergence order consuming an as small as possible computational cost. In addition, the theoretical advantages of the new method are based on the assumption that the Jacobian matrix is dense and that LU factorization is used to solve the systems with the Jacobian. This assumption is not correct for sparse Jacobian matrices.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **Method** & \(k\) & \(\left|\left|x^{(k)}-x^{(k-1)}\right|\right|\) & \(\left|\left|F(x^{(k)})\right|\right|\) & \(\rho\) \\ \hline \(F_{1}\) & NM & 8 & \(2.42128\times 10^{-192}\) & \(1.06480\times 10^{-383}\) & 1.99667 \\  & CM4 & 5 & \(5.59843\times 10^{-147}\) & \(2.69120\times 10^{-586}\) & 4.00129 \\  & SNAM & 4 & \(3.76810\times 10^{-39}\) & \(3.25655\times 10^{-227}\) & 6.09363 \\  & CHM & 4 & \(4.18959\times 10^{-123}\) & \(4.03125\times 10^{-736}\) & 5.99962 \\  & CTVM & 4 & \(2.07203\times 10^{-100}\) & \(2.63883\times 10^{-597}\) & 6.00033 \\  & M6 & 4 & \(7.65662\times 10^{-119}\) & \(1.55028\times 10^{-710}\) & 6.00589 \\ \hline \(F_{2}\) & NM & 9 & \(3.41596\times 10^{-116}\) & \(2.48971\times 10^{-232}\) & 1.97549 \\  & CM4 & 5 & \(3.73825\times 10^{-90}\) & \(1.20501\times 10^{-359}\) & 4.02761 \\  & SNAM & 4 & \(9.18821\times 10^{-35}\) & \(6.76819\times 10^{-207}\) & 5.98999 \\  & CHM & 4 & \(8.31995\times 10^{-52}\) & \(8.11818\times 10^{-310}\) & 5.72008 \\  & CTVM & 4 & \(3.82928\times 10^{-42}\) & \(4.59455\times 10^{-251}\) & 5.85429 \\  & M6 & 4 & \(8.13364\times 10^{-65}\) & \(6.14607\times 10^{-387}\) & 5.99644 \\ \hline \(F_{3}\) & NM & 22 & \(2.71070\times 10^{-196}\) & \(2.20459\times 10^{-392}\) & 1.99900 \\  & CM4 & 6 & \(2.26562\times 10^{-115}\) & \(1.03777\times 10^{-460}\) & 4.00061 \\  & SNAM & nc & & & \\  & CHM & 5 & \(2.79450\times 10^{-99}\) & \(4.68047\times 10^{-594}\) & 5.92903 \\  & CTVM & 5 & \(5.12075\times 10^{-193}\) & \(1.30600\times 10^{-1157}\) & 5.97091 \\  & M6 & 5 & \(1.99499\times 10^{-161}\) & \(3.41913\times 10^{-967}\) & 6.08153 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Numerical results for \(F_{i}(i=1,2,3)\) by the methods.  
**Acknowledgments:** This project is supported by the National Natural Science Foundation of China (Nos. 11547005 and 61572082), the PhD Start-up Fund of Liaoning Province of China (No. 201501196) and the Educational Commission Foundation of Liaoning Province of China (No. L2015012).  
**Author Contributions:** [PERSON] conceived and designed the experiments; [PERSON] performed the experiments; [PERSON] wrote the paper.  
**Conflicts of Interest:** The authors declare no conflict of interest.