Skip to main content
SpringerLink
Log in

Newton’s Method for M-Tensor Equations

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We are concerned with the tensor equations whose coefficient tensors are M-tensors. We first propose a Newton method for solving the equation with a positive constant term and establish its global and quadratic convergence. Then we extend the method to solve the equation with a nonnegative constant term and establish its convergence. At last, we do numerical experiments to test the proposed methods. The results show that the proposed methods are quite efficient.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability Statements

All data generated or analyzed during this study are included in this published article.

References

  1. Bader, B.W., Kolda, T.G., et al.: MATLAB tensor toolbox version 2.6 (2015)

  2. Bai, X., He, H., Ling, C., Zhou, G.: An nonnegativity preserving algorithm for multilinear systems with nonsingular M-tensors. Numer. Algorithms. 87(3), 1301–1320 (2021)

    Article  MathSciNet  Google Scholar 

  3. Berman, A., Plemmons, R.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadephia (1994)

    Book  Google Scholar 

  4. Brazell, M., Li, N., Navasca, C., Tamon, C.: Solving multilinear systems via tensor inversion. SIAM J. Matrix Anal. Appl. 34(2), 542–570 (2013)

    Article  MathSciNet  Google Scholar 

  5. Chang, K.C., Pearson, K., Zhang, T.: Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors. SIAM. J. Matrix Anal. Appl. 32(3), 806–819 (2011)

    Article  MathSciNet  Google Scholar 

  6. Ding, W., Qi, L., Wei, Y.: M-tensors and nonsingular M-tensors. Linear Algebra Appl. 439(10), 3264–3278 (2013)

    Article  MathSciNet  Google Scholar 

  7. Ding, W., Wei, Y.: Solving multi-linear systems with M-tensors. J. Sci. Comput. 68(2), 683–715 (2016)

    Article  MathSciNet  Google Scholar 

  8. Facchinei, F., Kanzow, C.: Beyond mootonicity in regularization methods for nonlinear complementarity problems. SIAM J. Control Optim. 37(4), 1150–1161 (1999)

    Article  MathSciNet  Google Scholar 

  9. Fan, H.Y., Zhang, L., Chu, E.K., Wei, Y.: Numerical solution to a linear equation with tensor product structure. Numer. Linear Algebra Appl. 24(6), e2106 (2017)

    Article  MathSciNet  Google Scholar 

  10. Gowda, M.S., Luo, Z., Qi, L., Xiu, N.: Z-tensor and complementarity problems (2015). arXiv:1510.07933

  11. Han, L.: A homotopy method for solving multilinear systems with M-tensors. Appl. Math. Lett. 69, 49–54 (2017)

    Article  MathSciNet  Google Scholar 

  12. He, H., Ling, C., Qi, L., Zhou, G.: A globally and quadratically convergent algorithm for solving multilinear systems with M-tensors. J. Sci. Comput. 76(3), 1718–1741 (2018)

    Article  MathSciNet  Google Scholar 

  13. Kressner, D., Tobler, C.: Krylov subspace methods for linear systems with tensor product structure. SIAM J. Matrix Anal. Appl. 31(4), 1688–1714 (2010)

    Article  MathSciNet  Google Scholar 

  14. Li, D.H., Guan, H.B., Xu, J.F.: Inexact Newton method for M-Tensor equations (2020). arXiv:2007.13324

  15. Li, D.H., Guan, H.B., Wang, X.Z.: Finding a nonnegative solution to an M-tensor equation. Pac. J. Optim. 16(3), 419–440 (2020)

    MathSciNet  MATH  Google Scholar 

  16. Li, D.H., Xie, S., Xu, H.R.: Splitting methods for tensor equations. Numer. Linear Algebra Appl. 24(5), e2102 (2017)

    Article  MathSciNet  Google Scholar 

  17. Li, Z., Dai, Y.-H., Gao, H.: Alternating projection method for a class of tensor equations. J. Comput. Appl. Math. 346, 490–504 (2019)

    Article  MathSciNet  Google Scholar 

  18. Li, X., Ng, M.K.: Solving sparse non-negative tensor equations: algorithms and applications. Front. Math. China 10(3), 649–680 (2015)

    Article  MathSciNet  Google Scholar 

  19. Liu, D., Li, W., Vong, S.W.: The tensor splitting with application to solve multi-linear systems. J. Comput. Appl. Math. 330, 75–94 (2018)

    Article  MathSciNet  Google Scholar 

  20. Lim, L.H.: Singular values and eigenvalues of tensors, a variational approach. In: Proceedings of the 1st IEEE International Workshop on Computational Advances of Multi-tensor Adaptive Processing, vol. 1, pp. 129–132 (2005)

  21. Lv, C.Q., Ma, C.F.: A Levenberg–Marquardt method for solving semi-symmetric tensor equations. J. Comput. Appl. Math. 332, 13–25 (2018)

    Article  MathSciNet  Google Scholar 

  22. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symbolic Comput. 40(6), 1302–1324 (2005)

    Article  MathSciNet  Google Scholar 

  23. Qi, L., Chen, H., Chen, Y.: Tensor Eigenvalues and Their Applications. Springer, Berlin (2018)

    Book  Google Scholar 

  24. Qi, L., Luo, Z.: Tensor Analysis. Spectral Theory and Special Tensors. SIAM, Philadelphia (2017)

    Book  Google Scholar 

  25. Wang, X., Che, M., Wei, Y.: Neural networks based approach solving multi-linear systems with M-tensors. Neural Comput. 351, 33–42 (2019)

    Google Scholar 

  26. Xie, Z.J., Jin, X.Q., Wei, Y.M.: Tensor methods for solving symmetric M-tensor systems. J. Sci. Comput. 74(1), 412–425 (2018)

    Article  MathSciNet  Google Scholar 

  27. Xie, Z.J., Jin, X.Q., Wei, Y.M.: A fast algorithm for solving circulant tensor systems. Linear Multilinear Algebra. 65(9), 1894–1904 (2017)

    Article  MathSciNet  Google Scholar 

  28. Xu, Y., Gu, W., Huang, Z.H.: Properties of the nonnegative solution set of multi-linear equations. Pac. J. Optim. 15(3), 441–456 (2019)

    MathSciNet  MATH  Google Scholar 

  29. Yan, W., Ling, C., Ling, L., He, H.: Generalized tensor equations with leading structured tensors. Appl. Math. Comput. 361, 311–324 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Zhang, L., Qi, L., Zhou, G.: M-tensor and some applications. SIAM J. Matrix Anal. Appl. 35(2), 437–452 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the NSF of China grant No.11771157, 11801184, the NSF of Guangdong Province grant No.2020B1515310013 and the Education Department of Hunan Province grant No.20C0559.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, Guangdong, China

    Dong-Hui Li, Jie-Feng Xu & Hong-Bo Guan

  2. School of Mathematics, Physics and Energy Engineering, Hunan Institute of Technology, HengYang, 421002, Hunan, China

    Hong-Bo Guan

Authors
  1. Dong-Hui Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jie-Feng Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hong-Bo Guan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hong-Bo Guan.

Additional information

Communicated by Liqun Qi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, DH., Xu, JF. & Guan, HB. Newton’s Method for M-Tensor Equations. J Optim Theory Appl 190, 628–649 (2021). https://doi.org/10.1007/s10957-021-01904-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-021-01904-0

Keywords

Mathematics Subject Classification

Search

Navigation