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New M-eigenvalue intervals and application to the strong ellipticity of fourth-order partially symmetric tensors
1. | School of Mathematics and Information Science, Weifang University, Weifang 261061, Shandong, China |
2. | School of Management Science, Qufu Normal University, Rizhao 276800, Shandong, China |
3. | Department of Mathematics and Statistics, Curtin University, Perth 6102, Western Australia, Australia |
M-eigenvalues of fourth-order partially symmetric tensors play an important role in nonlinear elasticity and materials. In this paper, we present some M-eigenvalue intervals to locate all M-eigenvalues of fourth-order partially symmetric tensors. It is proved that the new interval is tighter than the one proposed by He, Li and Wei [
References:
[1] |
H. Che, H. Chen and Y. Wang,
On the M-eigenvalue estimation of fourth-order partially symmetric tensors, Journal of Industrial and Management Optimization, 16 (2020), 309-324.
doi: 10.3934/jimo.2018153. |
[2] |
H. Che, H. Chen and Y. Wang, M-positive semi-definiteness and M-positive definiteness of fourth-order partially symmetric Cauchy tensors, Journal of Inequalities and Applications, (2019), Paper No. 32, 18 pp.
doi: 10.1186/s13660-019-1986-x. |
[3] |
H. Che, H. Chen and Y. Wang,
C-eigenvalue inclusion theorems for piezoelectric-type tensors, Applied Mathematics Letters, 89 (2019), 41-49.
doi: 10.1016/j.aml.2018.09.014. |
[4] |
H. Chen, Y. Wang and G. Zhou,
High-order sum-of-squares structured tensors: Theory and applications, Frontiers of Mathematics in China, 15 (2020), 255-284.
doi: 10.1007/s11464-020-0833-1. |
[5] |
H. Chen, Z. Huang and L. Qi,
Copositivity detection of tensors: Theory and algorithm, Journal of Optimization Theory and Applications, 174 (2017), 746-761.
doi: 10.1007/s10957-017-1131-2. |
[6] |
H. Chen, Y. Chen, G. Li and L. Qi, A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numerical Linear Algebra with Applications, 25 (2018), e2125, 16 pp.
doi: 10.1002/nla.2125. |
[7] |
H. Chen, Z. Huang and L. Qi,
Copositive tensor detection and its applications in physics and hypergraphs, Computational Optimization and Applications, 69 (2018), 133-158.
doi: 10.1007/s10589-017-9938-1. |
[8] |
H. Chen, L. Qi, Y. Wang and G. Zhou, Further results on sum-of-squares tensors, Optimization Methods and Software, (2020).
doi: 10.1080/10556788.2020.1768389. |
[9] |
H. Chen and Y. Wang,
On computing minimal H-eigenvalue of sign-structured tensors, Frontiers of Mathematics in China, 12 (2017), 1289-1302.
doi: 10.1007/s11464-017-0645-0. |
[10] |
H. Chen, L. Qi and Y. Song,
Column sufficient tensors and tensor complementarity problems, Frontiers of Mathematics in China, 13 (2018), 255-276.
doi: 10.1007/s11464-018-0681-4. |
[11] |
H. Chen, L. Qi, L. Caccetta and G. Zhou,
Birkhoff-von Neumann theorem and decomposition for doubly stochastic tensors, Linear Algebra and Its Applications, 583 (2019), 119-133.
doi: 10.1016/j.laa.2019.08.027. |
[12] |
S. Chirit$\check{a}$, A. Danescu and M. Ciarletta,
On the strong ellipticity of the anisotropic linearly elastic materials, Journal of Elasticity, 87 (2007), 1-27.
doi: 10.1007/s10659-006-9096-7. |
[13] |
B. Dacorogna,
Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete and Continuous Dynamical Systems, Series B, 1 (2001), 257-263.
doi: 10.3934/dcdsb.2001.1.257. |
[14] |
M. Dong and H. Chen, Geometry of the Copositive Tensor Cone and Its Dual, Asia-Pacific Journal of Operational Research, 2020. |
[15] |
D. Han, H. Dai and L. Qi,
Conditions for strong ellipticity of anisotropic elastic materials, Journal of Elasticity, 97 (2009), 1-13.
doi: 10.1007/s10659-009-9205-5. |
[16] |
J. He, C. Li and Y. Wei, M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity, Applied Mathematics Letters, 102 (2020), 106137.
doi: 10.1016/j.aml.2019.106137. |
[17] |
J. K. Knowles and E. Sternberg,
On the ellipticity of the equations of non-linear elastostatics for a special material, Journal of Elasticity, 5 (1975), 341-361.
doi: 10.1007/BF00126996. |
[18] |
J. K. Knowles and E. Sternberg,
On the failure of ellipticity of the equations for finite elastostatic plane strain, Archive for Rational Mechanics and Analysis, 63 (1976), 321-336.
doi: 10.1007/BF00279991. |
[19] |
S. Li, C. Li and Y. Li,
M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, Journal of Computational and Applied Mathematics, 356 (2019), 391-401.
doi: 10.1016/j.cam.2019.01.013. |
[20] |
C. Padovani,
Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.
doi: 10.1023/A:1020946506754. |
[21] |
L. Qi, H. Dai and D. Han,
Conditions for strong ellipticity and M-eigenvalues, Frontiers of Mathematics in China, 4 (2009), 349-364.
doi: 10.1007/s11464-009-0016-6. |
[22] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
[23] |
J. R. Walton and J. P. Wilber,
Sufficient conditions for strong ellipticity for a class of anisotropic materials, International Journal of Nonlinear Mechanics, 38 (2003), 441-455.
doi: 10.1016/S0020-7462(01)00066-X. |
[24] |
W. Wang, H. Chen and Y. Wang, A new C-eigenvalue interval for piezoelectric-type tensors, Applied Mathematics Letters, 100 (2020), 106035.
doi: 10.1016/j.aml.2019.106035. |
[25] |
C. Wang, H. Chen, Y. Wang and G. Zhou, On copositiveness identification of partially symmetric rectangular tensors, Journal of Computational and Applied Mathematics, 372 (2020), 112678.
doi: 10.1016/j.cam.2019.112678. |
[26] |
X. Wang, H. Chen and Y. Wang,
Solution structures of tensor complementarity problem, Frontiers of Mathematics in China, 13 (2018), 935-945.
doi: 10.1007/s11464-018-0675-2. |
[27] |
G. Wang, G. Zhou and L. Caccetta,
Z-eigenvalue inclusion theorems for tensors, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 187-198.
doi: 10.3934/dcdsb.2017009. |
[28] |
Y. Wang, L. Caccetta and G. Zhou,
Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numerical Linear Algebra and Applications, 22 (2015), 1059-1076.
doi: 10.1002/nla.1996. |
[29] |
Y. Wang, K. Zhang and H. Sun,
Criteria for strong H-tensors, Frontiers of Mathematics in China, 11 (2016), 577-592.
doi: 10.1007/s11464-016-0525-z. |
[30] |
Y. Wang, L. Qi and X. Zhang,
A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numerical Linear Algebra with Applications, 16 (2009), 589-601.
doi: 10.1002/nla.633. |
[31] |
K. Zhang and Y. Wang,
An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, Journal of Computational and Applied Mathematics, 305 (2016), 1-10.
doi: 10.1016/j.cam.2016.03.025. |
[32] |
G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numerical Linear Algebra with Applications, 25 (2018), e2134.
doi: 10.1002/nla.2134. |
show all references
References:
[1] |
H. Che, H. Chen and Y. Wang,
On the M-eigenvalue estimation of fourth-order partially symmetric tensors, Journal of Industrial and Management Optimization, 16 (2020), 309-324.
doi: 10.3934/jimo.2018153. |
[2] |
H. Che, H. Chen and Y. Wang, M-positive semi-definiteness and M-positive definiteness of fourth-order partially symmetric Cauchy tensors, Journal of Inequalities and Applications, (2019), Paper No. 32, 18 pp.
doi: 10.1186/s13660-019-1986-x. |
[3] |
H. Che, H. Chen and Y. Wang,
C-eigenvalue inclusion theorems for piezoelectric-type tensors, Applied Mathematics Letters, 89 (2019), 41-49.
doi: 10.1016/j.aml.2018.09.014. |
[4] |
H. Chen, Y. Wang and G. Zhou,
High-order sum-of-squares structured tensors: Theory and applications, Frontiers of Mathematics in China, 15 (2020), 255-284.
doi: 10.1007/s11464-020-0833-1. |
[5] |
H. Chen, Z. Huang and L. Qi,
Copositivity detection of tensors: Theory and algorithm, Journal of Optimization Theory and Applications, 174 (2017), 746-761.
doi: 10.1007/s10957-017-1131-2. |
[6] |
H. Chen, Y. Chen, G. Li and L. Qi, A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numerical Linear Algebra with Applications, 25 (2018), e2125, 16 pp.
doi: 10.1002/nla.2125. |
[7] |
H. Chen, Z. Huang and L. Qi,
Copositive tensor detection and its applications in physics and hypergraphs, Computational Optimization and Applications, 69 (2018), 133-158.
doi: 10.1007/s10589-017-9938-1. |
[8] |
H. Chen, L. Qi, Y. Wang and G. Zhou, Further results on sum-of-squares tensors, Optimization Methods and Software, (2020).
doi: 10.1080/10556788.2020.1768389. |
[9] |
H. Chen and Y. Wang,
On computing minimal H-eigenvalue of sign-structured tensors, Frontiers of Mathematics in China, 12 (2017), 1289-1302.
doi: 10.1007/s11464-017-0645-0. |
[10] |
H. Chen, L. Qi and Y. Song,
Column sufficient tensors and tensor complementarity problems, Frontiers of Mathematics in China, 13 (2018), 255-276.
doi: 10.1007/s11464-018-0681-4. |
[11] |
H. Chen, L. Qi, L. Caccetta and G. Zhou,
Birkhoff-von Neumann theorem and decomposition for doubly stochastic tensors, Linear Algebra and Its Applications, 583 (2019), 119-133.
doi: 10.1016/j.laa.2019.08.027. |
[12] |
S. Chirit$\check{a}$, A. Danescu and M. Ciarletta,
On the strong ellipticity of the anisotropic linearly elastic materials, Journal of Elasticity, 87 (2007), 1-27.
doi: 10.1007/s10659-006-9096-7. |
[13] |
B. Dacorogna,
Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete and Continuous Dynamical Systems, Series B, 1 (2001), 257-263.
doi: 10.3934/dcdsb.2001.1.257. |
[14] |
M. Dong and H. Chen, Geometry of the Copositive Tensor Cone and Its Dual, Asia-Pacific Journal of Operational Research, 2020. |
[15] |
D. Han, H. Dai and L. Qi,
Conditions for strong ellipticity of anisotropic elastic materials, Journal of Elasticity, 97 (2009), 1-13.
doi: 10.1007/s10659-009-9205-5. |
[16] |
J. He, C. Li and Y. Wei, M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity, Applied Mathematics Letters, 102 (2020), 106137.
doi: 10.1016/j.aml.2019.106137. |
[17] |
J. K. Knowles and E. Sternberg,
On the ellipticity of the equations of non-linear elastostatics for a special material, Journal of Elasticity, 5 (1975), 341-361.
doi: 10.1007/BF00126996. |
[18] |
J. K. Knowles and E. Sternberg,
On the failure of ellipticity of the equations for finite elastostatic plane strain, Archive for Rational Mechanics and Analysis, 63 (1976), 321-336.
doi: 10.1007/BF00279991. |
[19] |
S. Li, C. Li and Y. Li,
M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, Journal of Computational and Applied Mathematics, 356 (2019), 391-401.
doi: 10.1016/j.cam.2019.01.013. |
[20] |
C. Padovani,
Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.
doi: 10.1023/A:1020946506754. |
[21] |
L. Qi, H. Dai and D. Han,
Conditions for strong ellipticity and M-eigenvalues, Frontiers of Mathematics in China, 4 (2009), 349-364.
doi: 10.1007/s11464-009-0016-6. |
[22] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
[23] |
J. R. Walton and J. P. Wilber,
Sufficient conditions for strong ellipticity for a class of anisotropic materials, International Journal of Nonlinear Mechanics, 38 (2003), 441-455.
doi: 10.1016/S0020-7462(01)00066-X. |
[24] |
W. Wang, H. Chen and Y. Wang, A new C-eigenvalue interval for piezoelectric-type tensors, Applied Mathematics Letters, 100 (2020), 106035.
doi: 10.1016/j.aml.2019.106035. |
[25] |
C. Wang, H. Chen, Y. Wang and G. Zhou, On copositiveness identification of partially symmetric rectangular tensors, Journal of Computational and Applied Mathematics, 372 (2020), 112678.
doi: 10.1016/j.cam.2019.112678. |
[26] |
X. Wang, H. Chen and Y. Wang,
Solution structures of tensor complementarity problem, Frontiers of Mathematics in China, 13 (2018), 935-945.
doi: 10.1007/s11464-018-0675-2. |
[27] |
G. Wang, G. Zhou and L. Caccetta,
Z-eigenvalue inclusion theorems for tensors, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 187-198.
doi: 10.3934/dcdsb.2017009. |
[28] |
Y. Wang, L. Caccetta and G. Zhou,
Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numerical Linear Algebra and Applications, 22 (2015), 1059-1076.
doi: 10.1002/nla.1996. |
[29] |
Y. Wang, K. Zhang and H. Sun,
Criteria for strong H-tensors, Frontiers of Mathematics in China, 11 (2016), 577-592.
doi: 10.1007/s11464-016-0525-z. |
[30] |
Y. Wang, L. Qi and X. Zhang,
A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numerical Linear Algebra with Applications, 16 (2009), 589-601.
doi: 10.1002/nla.633. |
[31] |
K. Zhang and Y. Wang,
An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, Journal of Computational and Applied Mathematics, 305 (2016), 1-10.
doi: 10.1016/j.cam.2016.03.025. |
[32] |
G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numerical Linear Algebra with Applications, 25 (2018), e2134.
doi: 10.1002/nla.2134. |
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