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Numerical Algorithms

, Volume 80, Issue 3, pp 781–794 | Cite as

A new Brauer-type Z-eigenvalue inclusion set for tensors

  • Caili SangEmail author
Original Paper
  • 80 Downloads

Abstract

A new Brauer-type Z-eigenvalue inclusion set for tensors is given and proved to be tighter than those in Wang et al. (Discrete Contin. Dyn. Syst., Ser. B. 22(1), 187–198, 2017). Based on this set, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Numerical examples are reported to verify the theoretical results.

Keywords

Nonnegative tensors Z-eigenvalue Localization set Spectral radius Weakly symmetric 

Mathematics Subject Classification (2010)

15A18 15A42 15A69 

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Notes

Acknowledgments

The author is grateful to the referees, Prof. Yaotang Li, and Associate Prof. Chaoqian Li for their comments and suggestions.

Funding information

This work is supported by the National Natural Science Foundations of China (Grant No. 11501141), Foundation of Guizhou Science and Technology Department (Grant No. [2015]2073), and Natural Science Programs of Education Department of Guizhou Province (Grant No. [2016]066)

References

  1. 1.
    Qi, L.Q.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chang, K.C., Pearson, K.J., Zhang, T.: Some variational principles for Z-eigenvalues of nonnegative tensors. Linear Algebra Appl. 438, 4166–4182 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach. In: CAMSAP’05: Proceeding of the IEEE International Workshop on Computational Advances in Multisensor Adaptive Processing, vol. 1, pp. 129–132 (2005)Google Scholar
  4. 4.
    Lathauwer, L.D., Moor, B.D., Vandewalle, J.: On the best rank-1 and rank-(R 1, R 2,…,R N) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21, 1324–1342 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Zhang, T., Golub, G.H.: Rank-1 approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 23, 534–550 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kofidis, E., Regalia, R.: On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23, 863–884 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bloy, L., Verma, R.: On computing the underlying fiber directions from the diffusion orientation distribution function. In: Medical Image Computing and Computer-Assisted Intervention, vol. 5241, pp. 1–8. Springer (2008)Google Scholar
  8. 8.
    Qi, L.Q., Yu, G.H., Wu, E.X.: Higher order positive semi-definite diffusion tensor imaging. SIAM J. Matrix Anal. Appl. 3, 416–433 (2010)zbMATHGoogle Scholar
  9. 9.
    Kolda, T., Mayo, J.: Shifted power method for computing tensor eigenpairs. SIAM J. Matrix Anal. Appl. 32, 1095–1124 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Qi, L.: Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines. J. Symbolic Comput. 41, 1309–1327 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    DeVore, R.A., Temlyakov, V.N.: Some remarks on greedy algorithms. Adv. Comput. Math. 5, 173–187 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Falco, A., Nouy, A.: A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach. J. Math. Analysis and Appl. 376, 469–480 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wang, Y., Qi, L.: On the successive supersymmetric rank-1 decomposition of higher-order supersymmetric tensors. Numer. Linear Algebra Appl. 14, 503–519 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ammar, A., Chinesta, F., Falcó, A.: On the convergence of a greedy rank-one update algorithm for a class of linear systems. Arch. Comput. Methods Eng. 17, 473–486 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Qi, L.Q.: The best rank-one approximation ratio of a tensor space. SIAM J. Matrix Anal. Appl. 32, 430–442 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kofidis, E., Regalia, P.A.: On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23, 863–884 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Liu, Q.L., Li, Y.T.: Bounds for the Z-eigenpair of general nonnegative tensors. Open Math. 14(1), 181–194 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wang, G., Zhou, G.L., Caccetta, L.: Z-eigenvalue inclusion theorems for tensors. Discrete Contin. Dyn. Syst., Ser. B. 22(1), 187–198 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Song, Y.S., Qi, L.Q.: Spectral properties of positively homogeneous operators induced by higher order tensors. SIAM J. Matrix Anal. Appl. 34, 1581–1595 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Li, W., Liu, D.D., Vong, S.W.: Z-eigenpair bounds for an irreducible nonnegative tensor. Linear Algebra Appl. 483, 182–199 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    He, J.: Bounds for the largest eigenvalue of nonnegative tensors. J. Comput. Anal. Appl. 20(7), 1290–1301 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    He, J., Liu, Y.M., Ke, H., Tian, J.K., Li, X.: Bounds for the Z-spectral radius of nonnegative tensors. Springerplus. 5, 1727 (2016)CrossRefGoogle Scholar
  23. 23.
    Zhao, J.X.: A new Z-eigenvalue localization set for tensors. J. Inequal. Appl. 2017, 85 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    He, J., Huang, T.Z.: Upper bound for the largest Z-eigenvalue of positive tensors. Appl. Math. Lett. 38, 110–114 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Li, C.Q., Li, Y.T.: An eigenvalue localization set for tensor with applications to determine the positive (semi-)definiteness of tensors. Linear Multilinear Algebra. 64(4), 587–601 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.College of Data Science and Information EngineeringGuizhou Minzu UniversityGuiyangPeople’s Republic of China

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