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New techniques to develop higher order iterative methods for systems of nonlinear equations

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Abstract

In this paper, we discover new techniques to construct efficient higher order iterative methods for the longstanding problem of solving systems of nonlinear equations. Given any iterative method with the extended Newton iteration being a predictor, a new efficient higher order method can be constructed without more matrix inversions, but it requires introduction of a new function. In terms of our main theorems, we find that several well-known existing results are just special cases of ours. Applying the new techniques to some classical third or fourth order methods, we make light work of developing some new efficient higher order methods. For further discussion, we analyze the computational efficiencies of the newly developed methods, and make comprehensive comparisons between the new higher order methods and the corresponding lower order ones. Finally, we take several numerical experiments of solving nonlinear problems, and the numerical results are consistent with the theoretical analysis, to a large extent.

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Acknowledgements

This work is supported by National Nature Science Foundation of China with No. 12061048, and the Project of the Education Department of Jiangxi Province Research on the Teaching Reform with No. JXJG-20-1-16.

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  1. College of Mathematics and Computer, Xinyu University, 2666 Sunshine Avenue, Xinyu, 338004, China

    Xiao-Yong Xiao

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  1. Xiao-Yong Xiao
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Correspondence to Xiao-Yong Xiao.

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Communicated by Ke Chen.

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Xiao, XY. New techniques to develop higher order iterative methods for systems of nonlinear equations. Comp. Appl. Math. 41, 243 (2022). https://doi.org/10.1007/s40314-022-01959-3

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  • DOI: https://doi.org/10.1007/s40314-022-01959-3

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