High-order iterations for systems of nonlinear equations

Abstract

In this paper, several families of order $p(4\le p\le 6)$ for the solution of systems of nonlinear equations are developed and compared to existing methods. The necessary and sufficient conditions for pth order of convergence are given in terms of parameter matrices ${\tau }_k$ and ${\alpha }_k$. Several choices of parameter matrix ${\mathrm{\Theta }}_k$ determining ${\tau }_k$ are suggested. The proposed families include some well-known methods as particular cases. The comparison is made based on the total cost of an iteration and the CPU time.

Keywords:

• Systems of nonlinear equations
• Newton-type methods
• order of convergence

2010 Mathematics Subject Classifications:

• 65H10
• 65Y20
• 41A58

Acknowledgements

The authors wish to thank the editor and the anonymous referees for their valuable suggestions and comments on the first version of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported partially by the Foundation of Science and Technology of Mongolia [grant number SST_18/2018]. The research of the author Changbum Chun was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education [NRF-2016R1D1A1A09917373].