doi: 10.3934/jimo.2022077
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Parameterized S-type M-eigenvalue inclusion intervals for fourth-order partially symmetric tensors and its applications

1. 

School of Management Science, Qufu Normal University, Rizhao 276800, China

2. 

School of Mathematics and Information Science, Weifang University, Weifang 261061, China

*Corresponding author: Haitao Che

Received  October 2021 Revised  March 2022 Early access May 2022

Fund Project: This work was supported by the Natural Science Foundation of China (12071249, 12071250, 11901343), Shandong Provincial Natural Science Foundation (ZR2019MA022, ZR2020MA027, ZR2019PA016), Shandong Provincial Natural Science Foundation of Distinguished Young Scholars (ZR2021JQ01)

In this article, based on M-identity tensor, we establish some parameterized S-type inclusion intervals for fourth-order partially symmetric tensors. The new inclusion intervals are tighter than some existing results. Furthermore, some new S-type upper bounds on the M-spectral radius of fourth-order partially symmetric tensors are obtained. As applications, the new S-type upper bounds as the parameter of WQZ-algorithm can make the algorithm more rapidly converge to the largest M-eigenvalue of fourth-order partially symmetric tensors. Finally, we propose two sufficient conditions for the M-positive definiteness of fourth-order partially symmetric tensors.

Citation: Kaiping Liu, Haitao Che, Haibin Chen, Meixia Li. Parameterized S-type M-eigenvalue inclusion intervals for fourth-order partially symmetric tensors and its applications. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022077
References:
[1]

C. BuY. WeiL. Sun and J. Zhou, Brualdi-type eigenvalue inclusion sets of tensors, Linear Algebra Appl., 48 (2015), 168-175.  doi: 10.1016/j.laa.2015.04.034.

[2]

H. CheH. Chen and Y. Wang, On the M-eigenvalue estimation of fourthorder partially symmetric tensors, J. Ind. Manag. Optim., 16 (2020), 309-324.  doi: 10.3934/jimo.2018153.

[3]

H. CheH. Chen and Y. Wang, M-positive semi-definiteness and M-positive definiteness of fourth-order partially symmetric Cauchy tensors, J. Inequal. Appl., 2019 (2019), 1-18.  doi: 10.1186/s13660-019-1986-x.

[4]

H. ChenH. HeY. Wang and G. Zhou, An efficient alternating minimization method for fourth degree polynomial optimization, J. Global Optim., 82 (2022), 83-103.  doi: 10.1007/s10898-021-01060-9.

[5]

S. ChiritǎA. 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.

[6]

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

[7]

G. DahlJ. LeinaasJ. Myrheim and E. Ovrum, A tensor product matrix approximation problem in quantum physics, Linear Algebra Appl., 420 (2007), 711-725.  doi: 10.1016/j.laa.2006.08.026.

[8]

D. HanH. 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.

[9]

J. HeY. Liu and G. Xu, New S-type inclusion theorems for the M-eigenvalues of a 4th-order partially symmetric tensor with applications, Appl. Math. Comput., 398 (2021), 125992.  doi: 10.1016/j.amc.2021.125992.

[10]

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.

[11]

J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elasto-static plane strain, Arch. Rational Mech. Anal., 63 (1976), 321-333.  doi: 10.1007/BF00279991.

[12]

C. LiF. WangJ. ZhaoY. Zhu and Y. Li, Criterions for the positive definiteness of real supersymmetric tensors, J. Comput. Appl. Math., 255 (2014), 1-14.  doi: 10.1016/j.cam.2013.04.022.

[13]

S. LiC. Li and Y. Li, M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, J. Comput. Appl. Math., 365 (2019), 391-401.  doi: 10.1016/j.cam.2019.01.013.

[14]

C. LingJ. Nie and L. Qi, Bi-quadratic optimization over unit spheres and semidefinite programming relaxations, SIAM J. Optim., 20 (2009), 1286-1310.  doi: 10.1137/080729104.

[15]

L. Qi, The best rank-one approximation ratio of a tensor space, SIAM J. Matrix Anal. Appl., 32 (2011), 430-442.  doi: 10.1137/100795802.

[16]

L. QiH. 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.

[17]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, 2017. doi: 10.1137/1.9781611974751. ch1.

[18]

P. Vannucci, Anisotropic Elasticity, Springer, Singapore, 2018.

[19]

G. WangL. Sun and L. Liu, M-Eigenvalues-Based sufficient conditions for the positive definiteness of fourth-order partially symmetric tensors, Complexity, 3 (2020), 1-8. 

[20]

G. Wang, Y. Wang and Y. Wang, Some Ostrowski-type bound estimations of spectral radius for weakly irreducible nonnegative tensors, Linear and Multilinear Algebra, 68 (2020), 1817–-1834. doi: 10.1080/03081087.2018.1561823.

[21]

G. WangG. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.

[22]

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.

[23]

J. Zhao and C. Sang, An S-type upper bound for the largest singular value of nonnegative rectangular tensors, Open Math., 14 (2016), 925-933.  doi: 10.1515/math-2016-0085.

show all references

References:
[1]

C. BuY. WeiL. Sun and J. Zhou, Brualdi-type eigenvalue inclusion sets of tensors, Linear Algebra Appl., 48 (2015), 168-175.  doi: 10.1016/j.laa.2015.04.034.

[2]

H. CheH. Chen and Y. Wang, On the M-eigenvalue estimation of fourthorder partially symmetric tensors, J. Ind. Manag. Optim., 16 (2020), 309-324.  doi: 10.3934/jimo.2018153.

[3]

H. CheH. Chen and Y. Wang, M-positive semi-definiteness and M-positive definiteness of fourth-order partially symmetric Cauchy tensors, J. Inequal. Appl., 2019 (2019), 1-18.  doi: 10.1186/s13660-019-1986-x.

[4]

H. ChenH. HeY. Wang and G. Zhou, An efficient alternating minimization method for fourth degree polynomial optimization, J. Global Optim., 82 (2022), 83-103.  doi: 10.1007/s10898-021-01060-9.

[5]

S. ChiritǎA. 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.

[6]

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

[7]

G. DahlJ. LeinaasJ. Myrheim and E. Ovrum, A tensor product matrix approximation problem in quantum physics, Linear Algebra Appl., 420 (2007), 711-725.  doi: 10.1016/j.laa.2006.08.026.

[8]

D. HanH. 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.

[9]

J. HeY. Liu and G. Xu, New S-type inclusion theorems for the M-eigenvalues of a 4th-order partially symmetric tensor with applications, Appl. Math. Comput., 398 (2021), 125992.  doi: 10.1016/j.amc.2021.125992.

[10]

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.

[11]

J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elasto-static plane strain, Arch. Rational Mech. Anal., 63 (1976), 321-333.  doi: 10.1007/BF00279991.

[12]

C. LiF. WangJ. ZhaoY. Zhu and Y. Li, Criterions for the positive definiteness of real supersymmetric tensors, J. Comput. Appl. Math., 255 (2014), 1-14.  doi: 10.1016/j.cam.2013.04.022.

[13]

S. LiC. Li and Y. Li, M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, J. Comput. Appl. Math., 365 (2019), 391-401.  doi: 10.1016/j.cam.2019.01.013.

[14]

C. LingJ. Nie and L. Qi, Bi-quadratic optimization over unit spheres and semidefinite programming relaxations, SIAM J. Optim., 20 (2009), 1286-1310.  doi: 10.1137/080729104.

[15]

L. Qi, The best rank-one approximation ratio of a tensor space, SIAM J. Matrix Anal. Appl., 32 (2011), 430-442.  doi: 10.1137/100795802.

[16]

L. QiH. 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.

[17]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, 2017. doi: 10.1137/1.9781611974751. ch1.

[18]

P. Vannucci, Anisotropic Elasticity, Springer, Singapore, 2018.

[19]

G. WangL. Sun and L. Liu, M-Eigenvalues-Based sufficient conditions for the positive definiteness of fourth-order partially symmetric tensors, Complexity, 3 (2020), 1-8. 

[20]

G. Wang, Y. Wang and Y. Wang, Some Ostrowski-type bound estimations of spectral radius for weakly irreducible nonnegative tensors, Linear and Multilinear Algebra, 68 (2020), 1817–-1834. doi: 10.1080/03081087.2018.1561823.

[21]

G. WangG. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.

[22]

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.

[23]

J. Zhao and C. Sang, An S-type upper bound for the largest singular value of nonnegative rectangular tensors, Open Math., 14 (2016), 925-933.  doi: 10.1515/math-2016-0085.

Figure 1.  The comparison of $ \bar{\Gamma}(\mathscr{A}) $, $ \bar{\Delta}(\mathscr{A}) $, $ \Gamma(\mathscr{A}) $ and $ \Delta(\mathscr{A}) $
Figure 2.  Analysis of convergence of WQZ algorithm for the different parameters
Figure 3.  Numerical results for CaMg(CO$ _3 $)$ _2 $-dolomite elastic tensor
Table 1.  The comparison of inclusion intervals
Reference Inclusion interval
Lemma 1.1 [9] [-18, 18]
Lemma 1.2 [9] [-16.6119, 16.6119]
Theorem 2.1 [-14, 15]
Theorem 2.2 [-13, 12.2621]
Reference Inclusion interval
Lemma 1.1 [9] [-18, 18]
Lemma 1.2 [9] [-16.6119, 16.6119]
Theorem 2.1 [-14, 15]
Theorem 2.2 [-13, 12.2621]
Table 2.  The influence on the choose of parameter α, β
α=β [-2, 2]T [1, 3]T [3, 2]T
Theorem 2.2 [-15.0948, 14.5125] [-14.7703, 18.1980] [-15.0948, 20.4659]
α=β [-2, 2]T [1, 3]T [3, 2]T
Theorem 2.2 [-15.0948, 14.5125] [-14.7703, 18.1980] [-15.0948, 20.4659]
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