American Institute of Mathematical Sciences

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January  2020, 16(1): 309-324. doi: 10.3934/jimo.2018153

On the M-eigenvalue estimation of fourth-order partially symmetric tensors

Haitao Che 1,, , Haibin Chen 2, and  Yiju Wang 2,

1. 

School of Mathematics and Information Science, Weifang University, Weifang Shandong, 261061, China
2. 

School of Management Science, Qufu Normal University, Rizhao Shandong, 276826, China

* Corresponding author: Haitao Che

Received  January 2018 Revised  May 2018 Published  September 2018

Fund Project: This project is supported by the Natural Science Foundation of China (11401438, 11671228, 11601261, 11571120), Shandong Provincial Natural Science Foundation (ZR2016AQ12), Project of Shandong Province Higher Educational Science and Technology Program(Grant No. J14LI52), and China Postdoctoral Science Foundation (Grant No. 2017M622163, 2018T110669).

In this article, the M-eigenvalue of fourth-order partially symmetric tensors is estimated by choosing different components of M-eigenvector. As an application, some upper bounds for the M-spectral radius of nonnegative fourth-order partially symmetric tensors are discussed, which are sharper than existing upper bounds. Finally, numerical examples are reported to verify the obtained results.

Keywords: M-eigenvalue, Z-eigenvalue, fourth-order partially symmetric tensors, spectral property, localization theorem, nonnegative tensors.
Mathematics Subject Classification: Primary: 15A06, 74B20; Secondary: 47J25.
Citation: Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial & Management Optimization, 2020, 16 (1) : 309-324. doi: 10.3934/jimo.2018153
References:
[1]

K. Chang, K. Pearson and T. Zhang, Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182.  doi: 10.1016/j.laa.2013.02.013.  Google Scholar

[2]

H. Chen, Z. Huang and L. Qi, Copositivity detection of tensors: Theory and algorithm, J. Optimiz. Theory Appl., 174 (2017), 746-761.  doi: 10.1007/s10957-017-1131-2.  Google Scholar

[3]

H. Chen, Y. Chen, G. Li and L. Qi, A semi-definite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numer. Linear Algebra Appl., 25 (2018), e2125, 16pp. doi: 10.1002/nla.2125.  Google Scholar

[4]

H. Chen, Z. Huang and L. Qi, Copositive tensor detection and its applications in physics and hypergraphs, Comput. Optim. Appl., 69 (2018), 133-158.  doi: 10.1007/s10589-017-9938-1.  Google Scholar

[5]

H. Chen and Y. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Front. Math. China, 12 (2017), 1289-1302.  doi: 10.1007/s11464-017-0645-0.  Google Scholar

[6]

H. Chen, L. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Front. Math. China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.  Google Scholar

[7]

S. Chirit$\check{a}$, A. Danescu and M. Ciarletta, On the srtong ellipticity of the anisotropic linearly elastic materials, J. Elasticity, 87 (2007), 1-27.  doi: 10.1007/s10659-006-9096-7.  Google Scholar

[8]

B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete Cont. Dyn-B, 1 (2001), 257-263.  doi: 10.3934/dcdsb.2001.1.257.  Google Scholar

[9]

W. Ding, J. Liu, L. Qi and H. Yan, Elasticity M-tensors and the strong ellipticity condition, preprint, arXiv: 1705.09911. Google Scholar

[10]

D. Han, H. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, J. Elasticity, 97 (2009), 1-13.  doi: 10.1007/s10659-009-9205-5.  Google Scholar

[11]

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

[12]

Z. Huang and L. Qi, Positive definiteness of paired symmetric tensors and elasticity tensors, J. Comput. Appl. Math., 338 (2018), 22-43.  doi: 10.1016/j.cam.2018.01.025.  Google Scholar

[13]

J. K. Knowles and E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, J. Elasticity, 5 (1975), 341-361.  doi: 10.1007/BF00126996.  Google Scholar

[14]

J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal., 63 (1977), 321-336.  doi: 10.1007/BF00279991.  Google Scholar

[15]

E. Kofidis and P. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.  Google Scholar

[16]

C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.  doi: 10.1023/A:1020946506754.  Google Scholar

[17]

L. Qi, H. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Front. Math. China, 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.  Google Scholar

[18]

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

[19]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595.  doi: 10.1137/130909135.  Google Scholar

[20]

J. R. Walton and J. P. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, Int. J. Nonlin. Mech., 38 (2003), 441-455.  doi: 10.1016/S0020-7462(01)00066-X.  Google Scholar

[21]

Y. Wang, L. Caccetta and G. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear Algebra and Appl., 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.  Google Scholar

[22]

X. Wang, H. Chen and Y. Wang, Solution structures of tensor complementarity problem, Front. Math. China, 13 (2018), 935-945.  doi: 10.1007/s11464-018-0675-2.  Google Scholar

[23]

Y. Wang, L. 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

[24]

Y. Wang, K. Zhang and H. Sun, Criteria for strong H-tensors, Front. Mathe. China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.  Google Scholar

[25]

G. Wang, G. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete Cont. Dyn-B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.  Google Scholar

[26]

T. Zhang and G. Golub, Rank-1 approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.  Google Scholar

[27]

K. Zhang and Y. Wang, An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Comput. Appl. Math., 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.  Google Scholar

[28]

G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear Algebra Appl., 25 (2018), e2134, 10pp. doi: 10.1002/nla.2134.  Google Scholar

show all references

References:
[1]

K. Chang, K. Pearson and T. Zhang, Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182.  doi: 10.1016/j.laa.2013.02.013.  Google Scholar

[2]

H. Chen, Z. Huang and L. Qi, Copositivity detection of tensors: Theory and algorithm, J. Optimiz. Theory Appl., 174 (2017), 746-761.  doi: 10.1007/s10957-017-1131-2.  Google Scholar

[3]

H. Chen, Y. Chen, G. Li and L. Qi, A semi-definite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numer. Linear Algebra Appl., 25 (2018), e2125, 16pp. doi: 10.1002/nla.2125.  Google Scholar

[4]

H. Chen, Z. Huang and L. Qi, Copositive tensor detection and its applications in physics and hypergraphs, Comput. Optim. Appl., 69 (2018), 133-158.  doi: 10.1007/s10589-017-9938-1.  Google Scholar

[5]

H. Chen and Y. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Front. Math. China, 12 (2017), 1289-1302.  doi: 10.1007/s11464-017-0645-0.  Google Scholar

[6]

H. Chen, L. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Front. Math. China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.  Google Scholar

[7]

S. Chirit$\check{a}$, A. Danescu and M. Ciarletta, On the srtong ellipticity of the anisotropic linearly elastic materials, J. Elasticity, 87 (2007), 1-27.  doi: 10.1007/s10659-006-9096-7.  Google Scholar

[8]

B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete Cont. Dyn-B, 1 (2001), 257-263.  doi: 10.3934/dcdsb.2001.1.257.  Google Scholar

[9]

W. Ding, J. Liu, L. Qi and H. Yan, Elasticity M-tensors and the strong ellipticity condition, preprint, arXiv: 1705.09911. Google Scholar

[10]

D. Han, H. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, J. Elasticity, 97 (2009), 1-13.  doi: 10.1007/s10659-009-9205-5.  Google Scholar

[11]

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

[12]

Z. Huang and L. Qi, Positive definiteness of paired symmetric tensors and elasticity tensors, J. Comput. Appl. Math., 338 (2018), 22-43.  doi: 10.1016/j.cam.2018.01.025.  Google Scholar

[13]

J. K. Knowles and E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, J. Elasticity, 5 (1975), 341-361.  doi: 10.1007/BF00126996.  Google Scholar

[14]

J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal., 63 (1977), 321-336.  doi: 10.1007/BF00279991.  Google Scholar

[15]

E. Kofidis and P. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.  Google Scholar

[16]

C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.  doi: 10.1023/A:1020946506754.  Google Scholar

[17]

L. Qi, H. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Front. Math. China, 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.  Google Scholar

[18]

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

[19]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595.  doi: 10.1137/130909135.  Google Scholar

[20]

J. R. Walton and J. P. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, Int. J. Nonlin. Mech., 38 (2003), 441-455.  doi: 10.1016/S0020-7462(01)00066-X.  Google Scholar

[21]

Y. Wang, L. Caccetta and G. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear Algebra and Appl., 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.  Google Scholar

[22]

X. Wang, H. Chen and Y. Wang, Solution structures of tensor complementarity problem, Front. Math. China, 13 (2018), 935-945.  doi: 10.1007/s11464-018-0675-2.  Google Scholar

[23]

Y. Wang, L. 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

[24]

Y. Wang, K. Zhang and H. Sun, Criteria for strong H-tensors, Front. Mathe. China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.  Google Scholar

[25]

G. Wang, G. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete Cont. Dyn-B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.  Google Scholar

[26]

T. Zhang and G. Golub, Rank-1 approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.  Google Scholar

[27]

K. Zhang and Y. Wang, An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Comput. Appl. Math., 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.  Google Scholar

[28]

G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear Algebra Appl., 25 (2018), e2134, 10pp. doi: 10.1002/nla.2134.  Google Scholar

Figure 1.  The comparisons of $ \Gamma(\mathcal{C})$, $ \mathcal{L}(\mathcal{C})$, $ \mathcal{M}(\mathcal{C})$ and $ \mathcal{N}(\mathcal{C})$
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Figure 2.  The comparisons of $ \Gamma(\mathcal{C})$, $ \mathcal{L}(\mathcal{C})$ and $ \mathcal{M}(\mathcal{C})$
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Figure 3.  The comparisons of $ \Gamma(\mathcal{C})$, $ \mathcal{L}(\mathcal{C})$ and $ \mathcal{M}(\mathcal{C})$
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