Advertisement

Numerical Algorithms

, Volume 81, Issue 3, pp 1113–1128 | Cite as

Two non-parameter iterative algorithms for identifying strong \(\mathcal {H}\)-tensors

  • Yangyang Xu
  • Ruijuan Zhao
  • Bing ZhengEmail author
Original Paper
  • 55 Downloads

Abstract

The strong \(\mathcal {H}\)-tensors have important applications in many areas of science and engineering, e.g., the determination of positive definiteness for an even-order homogeneous polynomial form in the real field. In this paper, we propose two iterative algorithms with non-parameter for identifying strong \(\mathcal {H}\)-tensors, which overcome the drawback of choosing the best value of parameter 𝜖 in some existing algorithms given by Li et al. and Liu et al. (J. Comput. Appl. Math., 255, 1–14, 2014 and Comput. Appl. Math. 36, 1623–1635, 2017). Some numerical experiments are performed to illustrate the feasibility and effectiveness of our algorithms.

Keywords

Strong \(\mathcal {H}\)-tensors Non-parameter Iterative algorithm Positive definiteness 

Mathematics Subject Classification (2010)

15A18 15A69 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Funding information

This work was supported by the Natural Science Foundation of China (No. 11571004) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-it54).

References

  1. 1.
    Bose, N.K., Modarressi, A.R.: General procedure for multivariable polynomial positivity with control applications. IEEE Trans. Autom. Control 21, 696–701 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ding, W.Y., Qi, L.Q., Wei, Y.M.: \(\mathcal {M}\)-tensors and nonsingular \(\mathcal {M}\)-tensors. Linear Algebra Appl. 439, 3264–3278 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bose, N.K., Newcomb, R.W.: Tellegon’s theorem and multivariable realizability theory. Int. J. Electron. 36, 417–425 (1974)CrossRefGoogle Scholar
  4. 4.
    Bose, N.K., Kamat, P.S.: Algorithm for stability test of multidimensional filters. IEEE Trans. Acoust. Speech Signal Process. 22, 169–175 (1974)Google Scholar
  5. 5.
    Hu, S.L., Huang, Z.H., Qi, L.Q.: Strictly nonnegative tensors and nonnegative tensor partition. Sci. China Math. 57, 181–195 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bose, N.K.: Applied Multidimensional System Theory. Van Nostrand Rheinhold, New York (1982)zbMATHGoogle Scholar
  7. 7.
    Hasan, M.A., Hasan, A.A.: A procedure for the positive definiteness of forms of even-order. IEEE Trans. Autom. Control 41, 615–617 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hsu, J.C., Meyer, A.U.: Mordern Control Principles and Applications. McGraw-Hill, New York (1968)Google Scholar
  9. 9.
    Qi, L.Q., Song, Y.S.: An even order symmetric \(\mathcal {B}\)-tensor is positive definite. Linear Algebra Appl. 457, 303–312 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Anderson, B.D., Bose, N.K., Jury, E.I.: Output feedback stabilization and related problems-solutions via decision methods. IEEE Trans. Automat. Control 20, 55–66 (1975)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Li, Y.T., Liu, Q.L., Qi, L.Q.: Programmable criteria for strong \(\mathcal {H}\)-tensors. Numer. Algor. 74, 199–221 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ni, Q., Qi, L.Q., Wang, F.: An eigenvalue method for testing positive definiteness of a multivariate form. IEEE Trans. Autom. Control 53, 1096–1107 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Xu, Y.Y., Zhao, R.J., Zheng, B.: Some criteria for identifying strong \(\mathcal {H}\)-tensors. Numer. Algor. (2018).  https://doi.org/10.1007/s11075-018-0519-x
  14. 14.
    Wang, F., Sun, D.S.: New criteria for \(\mathcal {H}\)-tensors and an application. J. Inequal. Appl. 96, 1–12 (2016)MathSciNetGoogle Scholar
  15. 15.
    Wang, F., Sun, D.S., Zhao, J.X., Li, C.Q.: New practical criteria for \(\mathcal {H}\)-tensors and its application. Linear Multilinear Algebra 65, 269–283 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kannan, M.R., Shaked-Monderer, N., Berman, A.: Some properties of strong \(\mathcal {H}\)-tensors and general \(\mathcal {H}\)-tensors. Linear Algebra Appl. 476, 42–55 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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
  18. 18.
    Li, C.Q., Wang, F., Zhao, J.X., Li, Y.T.: Criterions for the positive definiteness of real supersymmetric tensors. J. Comput. Appl. Math. 255, 1–14 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Li, L., Niki, H., Sasanabe, M.: A nonparameter criterion for generalized diagonally dominant matrices. Int. J. Comput. Math. 71, 267–275 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chen, H.B., Qi, L.Q.: Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. J. Ind. Manag. Optim. 11, 1263–1274 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kohno, T., Niki, H., Sawami, H., Gao, Y.M.: An iterative test for H-matrix. J. Comput. Appl. Math. 115, 349–355 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach. In: CAMSAP’05: Proceeding of the IEEE Interational Workshop on Computational Advances in Multi-Sensor Adaptive Processing, vol. 1, pp 129–132 (2005)Google Scholar
  23. 23.
    Qi, L.Q.: Eigenvalues of a real supersymmetric tensor. J. Symbolic Comput. 40, 1302–1324 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liu, Q.L., Li, C.Q., Li, Y.T.: On the iterative criterion for strong \(\mathcal {H}\)-tensors. Comput. Appl. Math. 36, 1623–1635 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, X.Z., Wei, Y.M.: \(\mathcal {H}\)-tensors and nonsingular \(\mathcal {H}\)-tensors. Front. Math. China 11, 557–575 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wang, F., Sun, D.S.: New criteria for \(\mathcal {H}\)-tensors and an application. Open Math. 13, 609–616 (2015)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Zhao, R.J., Gao, L., Liu, Q.L., Li, Y.T.: Criterions for identifying \(\mathcal {H}\)-tensors. Front. Math. China 11, 661–678 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zhang, L.P., Qi, L.Q., Zhou, G.L.: \(\mathcal {M}\)-tensors and some applications. SIAM J. Matrix Anal. Appl. 35, 437–452 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Qi, L.Q., Luo, Z.Y.: Tensor Analysis: Spectral Theory and Special Tensors. SIAM, Philadelphia (2017)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China

Personalised recommendations