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Iterative methods for solving tensor equations based on exponential acceleration

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Abstract

The tensor equation \(\mathcal {A}{\varvec{x}}^{m-1}={\varvec{b}}\) with the tensor \(\mathcal {A}\) of order m and dimension n and the vector \({\varvec{b}}\), has practical applications in several fields including signal processing, high-dimensional PDEs, high-order statistics, and so on. In this paper, a class of exponential accelerated iterative methods is proposed for solving the tensor equation mentioned above in the sense that the coefficient tensor \(\mathcal {A}\) is a symmetric and nonsingular or singular \(\mathcal {M}\)-tensor. The obtained iterative schemes involve the classical Newton’s method as a special case. It is shown that the proposed method for nonsingular case is superlinearly convergent, while for singular cases, it is linearly convergent. The performed numerical experiments demonstrate that our methods outperform some existing ones.

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Acknowledgements

The authors are thankful to the handling editor and the referees for their constructive comments and suggestions, which greatly improve the quality of this paper.

Funding

This work was supported by National Natural Science Foundation of China (Nos. 11961057, 12201267, 12361081), the Natural Science Foundation of Gansu Province (Nos. 21JR1RE287, 22JR5RA559) the Innovation Foundation of Education Department of Gansu Province (Nos. 2021B-221, 2023B-135), and the Science Foundation of Tianshui Normal University (No. CXJ2021-01).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Tianshui Normal University, Tianshui, 741001, People’s Republic of China

    Maolin Liang & Lifang Dai

  2. School of Information Engineering, Lanzhou University of Finance and Economics, Lanzhou, 730101, People’s Republic of China

    Ruijuan Zhao

Authors
  1. Maolin Liang
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  2. Lifang Dai
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Contributions

M. Liang provided the methodology along with the problem under consideration and re-editing and correcting the manuscript, L. Dai implemented the scheme and edited the manuscript, and R. Zhao completed the numerical experiments of the related algorithms.

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Correspondence to Maolin Liang.

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Liang, M., Dai, L. & Zhao, R. Iterative methods for solving tensor equations based on exponential acceleration. Numer Algor (2023). https://doi.org/10.1007/s11075-023-01692-w

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