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A new iterative refinement for ill-conditioned linear systems based on discrete gradient

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Abstract

In this paper, a new iterative refinement for ill-conditioned linear systems is derived based on discrete gradient methods for gradient systems. It is proved that the new method is convergent for any initial values irrespective of the choice of the stepsize h. Some properties of the new iterative refinement are presented. It is shown that the condition number of the coefficient matrix in the linear system to be solved in every step can be improved significantly compared with Wilkinson’s iterative refinement. The numerical experiments illustrate that the new method is more effective and efficient than Wilkinson’s iterative refinement when dealing with ill-conditioned linear systems.

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Acknowledgements

The authors sincerely thank the editors and referees for their kind and valuable comments of revision which improved the presentation of the paper.

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Authors and Affiliations

  1. College of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, 210023, People’s Republic of China

    Kai Liu & Jie Yang

  2. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, People’s Republic of China

    Changying Liu

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  1. Kai Liu
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  2. Jie Yang
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Correspondence to Jie Yang.

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The research was supported in part by the Natural Science Foundation of China under Grant 11701271 and by the Natural Science Foundation of the Jiangsu Higher Education Institutions under Grant 16KJB110010.

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Liu, K., Yang, J. & Liu, C. A new iterative refinement for ill-conditioned linear systems based on discrete gradient. Japan J. Indust. Appl. Math. 37, 803–818 (2020). https://doi.org/10.1007/s13160-020-00417-z

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  • DOI: https://doi.org/10.1007/s13160-020-00417-z

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