doi: 10.3934/jimo.2021176
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Bounds for spectral radius of nonnegative tensors using matrix-digragh-based approach

School of Mathematics and Statistics, Beihua University, Jilin, China

* Corresponding author: Hongbin Lv

Received  April 2021 Revised  August 2021 Early access October 2021

Fund Project: This work is supported by the Natural Sciences Program of Science and Technology of Jilin Province of China(20190201139JC)

We obtain the improved results of the upper and lower bounds for the spectral radius of a nonnegative tensor by its majorization matrix's digraph. Numerical examples are also given to show that our results are significantly superior to the results of related literature.

Citation: Guimin Liu, Hongbin Lv. Bounds for spectral radius of nonnegative tensors using matrix-digragh-based approach. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021176
References:
[1]

K. C. ChangK. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.  doi: 10.4310/CMS.2008.v6.n2.a12.  Google Scholar

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S. FriedlandS. Gaubert and L. Han, Peerron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl., 438 (2013), 738-749.  doi: 10.1016/j.laa.2011.02.042.  Google Scholar

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J. He and T. Huang, Upper bound for the lagest z-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.  Google Scholar

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S. L. HuZ. H. HuangC. Ling and L. Qi, On determinants and eigenvalue theory of tensors, J. Symbolic Comput., 50 (2013), 508-531.  doi: 10.1016/j.jsc.2012.10.001.  Google Scholar

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L. H. Lim, Singular values and eigenvalues of tensors: A Variational approach, CAMSAP '05: Proceeding of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, (2005), 129–132. Google Scholar

[6]

C. LiY. WangJ. Yi and Y. Li, Bounds for the spectral radius of nonnegative tensor, J. Ind. Manag. Optim., 12 (2016), 975-990.  doi: 10.3934/jimo.2016.12.975.  Google Scholar

[7]

L. Li and C. Li, New bounds for the spectral radius for nonnegative tensors, J. Inequal. Appl., 166 (2015), 1-9.  doi: 10.1186/s13660-015-0689-1.  Google Scholar

[8]

W. Li and M. Ng, Some bounds for the spectral radius of nonnegative tensors, Numer. Math., 130 (2015), 315-335.  doi: 10.1007/s00211-014-0666-5.  Google Scholar

[9]

M. NgL. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl, 31 (2009), 1090-1099.  doi: 10.1137/09074838X.  Google Scholar

[10]

K. J. Pearson, Essentially positive tensors, Int. J. Algebra, 4 (2010), 421-427.   Google Scholar

[11]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.  Google Scholar

[12]

L. QiW. Sun and Y. Wang, Numerical multilinear algebra and its applications, Front. Math. China, 2 (2007), 501-526.  doi: 10.1007/s11464-007-0031-4.  Google Scholar

[13]

L. Qi, Eigenvalues and invariants of tensors, J. Math. Anal. Appl., 325 (2007), 1363-1377.  doi: 10.1016/j.jmaa.2006.02.071.  Google Scholar

[14]

L. QiY. Wang and E. Wu, D-eigenvalues of diffusion kurtosis tensors, J. Comput. Appl. Math., 221 (2008), 150-157.  doi: 10.1016/j.cam.2007.10.012.  Google Scholar

[15] A. Roger and R. Charles, Matrix Analysis, The People's Posts and Telecommunications Press, 2007.   Google Scholar
[16]

T. Schultz and H. Seidel, Estimating crossing fibers: A tensor decomposition approach, IEEE Transactions on Visualization and Computer Graphics, 14 (2008), 1635-1642.  doi: 10.1109/TVCG.2008.128.  Google Scholar

[17]

Y. WangL. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.  doi: 10.1002/nla.633.  Google Scholar

[18]

Y. Yang and Q. Yang, Further results for perron-frobenius theorem for nonnegative tensors, SIAM. J. Matrix Anal. Appl., 31 (2010), 2517-2530.  doi: 10.1137/090778766.  Google Scholar

show all references

References:
[1]

K. C. ChangK. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.  doi: 10.4310/CMS.2008.v6.n2.a12.  Google Scholar

[2]

S. FriedlandS. Gaubert and L. Han, Peerron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl., 438 (2013), 738-749.  doi: 10.1016/j.laa.2011.02.042.  Google Scholar

[3]

J. He and T. Huang, Upper bound for the lagest z-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.  Google Scholar

[4]

S. L. HuZ. H. HuangC. Ling and L. Qi, On determinants and eigenvalue theory of tensors, J. Symbolic Comput., 50 (2013), 508-531.  doi: 10.1016/j.jsc.2012.10.001.  Google Scholar

[5]

L. H. Lim, Singular values and eigenvalues of tensors: A Variational approach, CAMSAP '05: Proceeding of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, (2005), 129–132. Google Scholar

[6]

C. LiY. WangJ. Yi and Y. Li, Bounds for the spectral radius of nonnegative tensor, J. Ind. Manag. Optim., 12 (2016), 975-990.  doi: 10.3934/jimo.2016.12.975.  Google Scholar

[7]

L. Li and C. Li, New bounds for the spectral radius for nonnegative tensors, J. Inequal. Appl., 166 (2015), 1-9.  doi: 10.1186/s13660-015-0689-1.  Google Scholar

[8]

W. Li and M. Ng, Some bounds for the spectral radius of nonnegative tensors, Numer. Math., 130 (2015), 315-335.  doi: 10.1007/s00211-014-0666-5.  Google Scholar

[9]

M. NgL. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl, 31 (2009), 1090-1099.  doi: 10.1137/09074838X.  Google Scholar

[10]

K. J. Pearson, Essentially positive tensors, Int. J. Algebra, 4 (2010), 421-427.   Google Scholar

[11]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.  Google Scholar

[12]

L. QiW. Sun and Y. Wang, Numerical multilinear algebra and its applications, Front. Math. China, 2 (2007), 501-526.  doi: 10.1007/s11464-007-0031-4.  Google Scholar

[13]

L. Qi, Eigenvalues and invariants of tensors, J. Math. Anal. Appl., 325 (2007), 1363-1377.  doi: 10.1016/j.jmaa.2006.02.071.  Google Scholar

[14]

L. QiY. Wang and E. Wu, D-eigenvalues of diffusion kurtosis tensors, J. Comput. Appl. Math., 221 (2008), 150-157.  doi: 10.1016/j.cam.2007.10.012.  Google Scholar

[15] A. Roger and R. Charles, Matrix Analysis, The People's Posts and Telecommunications Press, 2007.   Google Scholar
[16]

T. Schultz and H. Seidel, Estimating crossing fibers: A tensor decomposition approach, IEEE Transactions on Visualization and Computer Graphics, 14 (2008), 1635-1642.  doi: 10.1109/TVCG.2008.128.  Google Scholar

[17]

Y. WangL. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.  doi: 10.1002/nla.633.  Google Scholar

[18]

Y. Yang and Q. Yang, Further results for perron-frobenius theorem for nonnegative tensors, SIAM. J. Matrix Anal. Appl., 31 (2010), 2517-2530.  doi: 10.1137/090778766.  Google Scholar

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