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A two-step Broyden-like method for nonlinear equations

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Abstract

In this paper, based on a nonmonotone derivative-free line search, we propose a two-step Broyden-like method (denoted by TS-BLM) for solving the nonlinear equations. TS-BLM computes one quasi-Newton step at the beginning iterations. When the iterations are close to the solution set of the nonlinear equations, an additional approximate quasi-Newton step is calculated by solving a linear system formed by using the previous Broyden-like matrix. We prove that TS-BLM converges globally under appropriate conditions. Moreover, we analyze the convergence rate of TS-BLM depending on the approximation of the Broyden-like matrix to the Jacobian. Some numerical results are reported to show the superior numerical performances of TS-BLM compared with the traditional BLM.

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The data sets generated during the current study are available from the authors on reasonable request.

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Acknowledgements

The authors would like to thank two referees for their valuable suggestions that greatly improved the paper.

Funding

This work was partially supported by the National Natural Science Foundation of China (Grant No. 12371305), the Natural Science Foundation of Shandong Province (Grant No. ZR2023MA020), the Natural Science Foundation of Henan Province (Grant No. 222300420520) and the Key Scientific Research Projects of Higher Education of Henan Province (Grant No. 22A110020).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China

    Jingyong Tang

  2. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, China

    Jinchuan Zhou

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  1. Jingyong Tang
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  2. Jinchuan Zhou
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Contributions

Jingyong Tang contributed to the analysis of the algorithm and manuscript preparation; Jinchuan Zhou helped perform the analysis with constructive discussions.

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Correspondence to Jingyong Tang.

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Tang, J., Zhou, J. A two-step Broyden-like method for nonlinear equations. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01827-7

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