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A modified Levenberg–Marquardt method with line search for nonlinear equations

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Abstract

In this paper, we propose a new modified Levenberg–Marquardt method for nonlinear equations. At every iteration, not only a general LM step, but also two additional approximate LM steps which save the Jacobian calculation and employ line search for the step size, are computed. Under the error bound condition which is weaker than nonsingularity, this method is shown to be of fourth convergence order. Numerical results show that the new method is very efficient and could save many calculations of the Jacobian.

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Acknowledgments

The author would like to thank Professor Yangfeng Su at Fudan University, China, and the referees for the useful comments and suggestions on the paper. The work is supported by the NSFC (61300048, 11371243), the Anhui Provincial Natural Science Foundation (1308085QF117, 1508085MA14), the Key Natural Science Foundation of Universities of Anhui Province (KJ2014A003, KJ2014A223), the major teaching reform project of Anhui higher education revitalization plan (2014ZDJY058) and the Excellent Young Talents in Universities of Anhui Province.

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  1. School of Mathematical Sciences, Huaibei Normal University, Huaibei, 235000, Anhui, People’s Republic of China

    Liang Chen

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  1. Liang Chen
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Chen, L. A modified Levenberg–Marquardt method with line search for nonlinear equations. Comput Optim Appl 65, 753–779 (2016). https://doi.org/10.1007/s10589-016-9852-y

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