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A new Jacobi-type iteration method for solving M-matrix or nonnegative linear systems

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Abstract

In this paper, based on the exponential integrator, a new Jacobi-type iteration method is proposed for solving linear system \(Ax=b\). The traditional Jacobi iteration method can be viewed as a special case of the new method. The convergence and two comparison theorems of the new Jacobi-type method are established for linear system with different type of coefficient matrices. The convergence of the traditional Jacobi iteration method follows immediately from these results. It is shown that for the linear system with coefficient matrix that is M-matrix or nonnegative matrix, the new method is convergent. Under suitable conditions, the spectral radius of iteration matrix for new method is much smaller than traditional Jacobi method for the case of nonnegative coefficient matrix. Numerical experiments are carried out to show the effectiveness of the new method when dealing with the nonnegative matrix system.

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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, Mingqian Zhang & Jie Yang

  2. College of Mathematical Sciences, Nanjing Tech University, Nanjing, 211816, People’s Republic of China

    Wei Shi

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  1. Kai Liu
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  2. Mingqian Zhang
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  3. Wei Shi
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  4. Jie Yang
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Correspondence to Wei Shi.

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The research was supported in part by the Natural Science Foundation of China under Grant 11701271 and by Jiangsu Qinglan Project.

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Liu, K., Zhang, M., Shi, W. et al. A new Jacobi-type iteration method for solving M-matrix or nonnegative linear systems. Japan J. Indust. Appl. Math. 39, 403–417 (2022). https://doi.org/10.1007/s13160-021-00488-6

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  • DOI: https://doi.org/10.1007/s13160-021-00488-6

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