Abstract
In this paper, an iterative method is proposed for finding the spectral radius of an irreducible nonnegative tensor based on diagonal similar tensors. The iterative method is convergent for irreducible nonnegative tensors and has an explicit linear convergence rate for generalized weakly positive tensors. Some numerical results are provided to illustrate the efficiency of the iterative method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benson AR (2019) Three hypergraph eigenvector centralities. SIAM J Math Data Sci 1:293–312
Brauer A (1957) A method for the computation of the greatest root of a positive matrix. J Soc Indust Appl Math 5:250–253
Bulò SR, Pelillo M (2009) New bounds on the clique number of graphs based on spectral hypergraph theory. In: Stützle T (ed) Learning and intelligent optimization. Springer, Berlin
Chang KC, Pearson KJ, Zhang T (2008) Perron-Frobenius theorem for nonnegative tensors. Commun Math Sci 6:507–520
Chang KC, Pearson KJ, Zhang T (2011) Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors. SIAM J Matrix Anal Appl 32:806–819
Chen L, Han L, Yin H, Zhou L (2019) A homotopy method for computing the largest eigenvalue of an irreducible nonnegative tensor. J Comput Appl Math 355:174–181
Friedland S, Gaubert S, Han L (2013) Perron-Frobenius theorem for nonnegative multilinear forms and extensions. Linear Algebra Appl 438:738–749
Hall C, Porsching T (1968) Computing the maximal eigenvalue and eigenvector of a nonnegative irreducible matrix. SIAM J Numer Anal 5:470–474
Lim LH (2005) Singular values and eigenvalues of tensors: a variational approach. In: Proceedings 1st IEEE International Workshop on Computational Advances of Multitensor Adaptive Processing, pp 129–132 and Tomorrow
Liu Y, Zhou G, Ibrahim NF (2010) An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor. J Comput Appl Math 235:286–292
Ng M, Qi L, Zhou G (2009) Finding the largest eigenvalue of a nonnegative tensor. SIAM J Matrix Anal Appl 31:1090–1099
Ni Q, Qi L, Wang F (2008) An eigenvalue method for testing the positive definiteness of a multivariant form. IEEE Trans Automat Contr 53:1096–1107
Pearson KJ (2010) Essentially positive tensors. Int J Algebra 4:421–427
Qi L (2005) Eigenvalues of a real supersymmetric tensor. J Symb Comput 40:1302–1324
Shao J (2013) A general product of tensors with applications. Linear Algebra Appl 439:2350–2366
Yang Y, Yang Q (2010) Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J Matrix Anal Appl 31:2517–2530
Yin J, Kong X, Zheng N (2015) Aitken extrapolation method for computing the largest eigenvalue of nonnegative tensors. Appl Math Comput 258:350–357
Zhang L, Qi L (2012) Linear convergence of an algorithm for computing the largest eigenvalue of a nonnegative tensor. Numer Linear Algebra Appl 19:830–841
Zhang L, Qi L, Xu Y (2012) Linear convergence of the LZI algorithm for weakly positive tensors. J Comput Math 30:24–33
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11801115, No. 12071097 and No. 12042103), the Natural Science Foundation of the Heilongjiang Province (No. QC2018002), the Fundamental Research Funds for the Central Universities, and the Natural Science Foundation of Inner Mongolia Higher Education (No. NJZY20120).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Zhong-Zhi Bai.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, J., Bu, C. An iterative method for finding the spectral radius of an irreducible nonnegative tensor. Comp. Appl. Math. 40, 8 (2021). https://doi.org/10.1007/s40314-020-01375-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-020-01375-5