The tensor splitting methods for solving tensor absolute value equation

Abstract

Recently, D.-D. Liu et al. (2018) presented the tensor splitting methods for solving multilinear systems and S.-Q. Du et al. (2018) generalized tensor absolute value equations. In this paper, we verify the existence of solutions of tensor absolute value equations and propose the tensor splitting methods for solving this class of equation. Furthermore, the convergence analysis of the tensor splitting method is also studied under suitable conditions. Finally, numerical examples show that our algorithm is an efficient iterative method.

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

References

  1. Chang K-C, Pearson K, Zhang T (2008) Perron-Frobenius theorem for nonnegative tensors. Commun. Math. Sci. 6:507–520

    Article  Google Scholar 

  2. Che M, Qi L, Wei Y (2016) Positive-definite tensors to nonlinear complementarity problems. J. Optim. Theory Appl. 168:475–487

    Article  Google Scholar 

  3. Che M, Qi L, Wei Y (2019) Stochastic \(R_0\) tensors to stochastic tensor complementarity problems. Optimization Letters 13:261–279

    Article  Google Scholar 

  4. Ding W-Y, Wei Y (2016) Solving multi-linear system with M-tensors. J. Sci. Comput. 68(2):689–715

    Article  Google Scholar 

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

    Article  Google Scholar 

  6. Du S-Q, Zhang L-P, Chen C-Y, Qi L-Q (2018) Tensor absolute value equations. Sci. China Math. https://doi.org/10.1007/s11425-017-9238-6

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  9. Li D-H, Xie S-L, Xu H-R (2017) Splitting methods for tensor equations. Numer. Linear Algebra Appl. https://doi.org/10.1002/nla.2102

    Article  MATH  Google Scholar 

  10. Li W, Liu D-D, Vong SW (2018) Comparison results for splitting iterations for solving multi-linear systems. Appl. Numer. Math. 134:105–121

    Article  Google Scholar 

  11. Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach, in: Proceedings of the IEEE International Workshop on Computational-Advances in Multi-Sensor Adaptive Processing, CAMSAP 05, vol.1, IEEE Computer Society Press, Piscataway, NJ, (2005), pp. 129–132

  12. Liu W, Li W (2016) On the inverse of a tensor. Linear Algebra Appl. 495:199–205

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  15. Mangasarian OL, Meyer RR (2006) Absolute value equations. Linear Algebra Appl. 419:359–367

    Article  Google Scholar 

  16. Qi L-Q (2005) Eigenvalues of a real supersymmetric tensor. J. Symbolic Comput. 40:1302–1324

    Article  Google Scholar 

  17. Varga RS (1962) Matrix Iterative Analysis. Prentice Hall, Englewood Cliffs, New Jersy

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  20. Zhang L-P, Qi L-Q, Zhou G-G (2014) M-tensors and some applications. SIAM J. Matrix Anal. Appl. 35:437–452

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Chang-Feng Ma.

Additional information

Publisher's Note

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

Communicated by Jinyun Yuan.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bu, F., Ma, C. The tensor splitting methods for solving tensor absolute value equation. Comp. Appl. Math. 39, 178 (2020). https://doi.org/10.1007/s40314-020-01195-7

Download citation

Keywords

  • Tensor absolute value equation
  • Tensor splitting
  • Strong \(\mathcal {M}\)-tensor

Mathematics Subject Classification

  • 15A69
  • 15A72
  • 53A45