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Robust sensitivity analysis for linear programming with ellipsoidal perturbation
$ E $-eigenvalue localization sets for tensors
1. | School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou 550025, China |
2. | College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, China |
Several existing $Z$-eigenvalue localization sets for tensors are first generalized to $E$-eigenvalue localization sets. And then two tighter $ E$-eigenvalue localization sets for tensors are presented. As applications, a sufficient condition for the positive definiteness of fourth-order real symmetric tensors, a sufficient condition for the positive semi-definiteness of fourth-order real symmetric tensors, and a new upper bound for the $ Z$-spectral radius of weakly symmetric nonnegative tensors are obtained. Finally, numerical examples are given to verify the theoretical results.
References:
[1] |
A. Ammar, F. Chinesta and A. Falcó,
On the convergence of a greedy rank-one update algorithm for a class of linear systems, Arch. Comput. Methods Eng., 17 (2010), 473-486.
doi: 10.1007/s11831-010-9048-z. |
[2] |
B. D. Anderson, N. K. Bose and E. I. Jury,
Output feedback stabilization and related problems-solutions via decision methods, IEEE Trans. Automat. Control, AC20 (1975), 53-66.
doi: 10.1109/tac.1975.1100846. |
[3] |
L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, Med. Image Comput. Comput. Assist. Interv., 5241 (2008), 1–8. Available from: https://www.ncbi.nlm.nih.gov/pubmed/18979725.
doi: 10.1007/978-3-540-85988-8_1. |
[4] |
N. K. Bose and P. S. Kamt,
Algorithm for stability test of multidimensional filters, IEEE Trans. Acoust. Speech Signal Process, 22 (1974), 307-314.
doi: 10.1109/TASSP.1974.1162592. |
[5] |
N. K. Bose and R. W. Newcomb,
Tellegon's theorem and multivariate realizability theory, Int. J. Electron, 36 (1974), 417-425.
doi: 10.1080/00207217408900421. |
[6] |
K. C. Chang, K. J. 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. |
[7] |
R. A. Devore and V. N. Temlyakov,
Some remarks on greedy algorithms, Adv. comput. Math., 5 (1996), 173-187.
doi: 10.1007/BF02124742. |
[8] |
A. Falco and A. Nouy,
A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach, J. Math. Anal. Appl., 376 (2011), 469-480.
doi: 10.1016/j.jmaa.2010.12.003. |
[9] |
J. He,
Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20 (2016), 1290-1301.
|
[10] |
J. He, Y. M. Liu, H. Ke, J. K. Tian and X. Li,
Bounds for the $Z$-spectral radius of nonnegative tensors, Springerplus, 5 (2016), 1727.
doi: 10.1186/s40064-016-3338-3. |
[11] |
J. He and T. Z. 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. |
[12] |
J. C. Hsu and A. U. Meyer, Modern Control Principles and Applications, The McGraw-Hill Series in Advanced Chemistry McGraw-Hill Book Co., Inc., New York-Toronto-London, 1956. |
[13] |
E. Kofidis and P. A. 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. |
[14] |
T. G. Kolda and J. R. Mayo,
Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095-1124.
doi: 10.1137/100801482. |
[15] |
L. D. Lathauwer, B. D. Moor and J. Vandewalle,
On the best rank-1 and rank-($R_1,R_2,\ldots ,R_N $) approximation of higer-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.
doi: 10.1137/S0895479898346995. |
[16] |
W. Li, D. Liu and S. W. Vong,
$Z$-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl., 483 (2015), 182-199.
doi: 10.1016/j.laa.2015.05.033. |
[17] |
C. Li and Y. Li,
An eigenvalue localization set for tensors with applications to determine the positive (semi-)definitenss of tensors, Linear Multilinear Algebra, 64 (2016), 587-601.
doi: 10.1080/03081087.2015.1049582. |
[18] |
C. Li, Y. Li and X. Kong,
New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 21 (2014), 39-50.
doi: 10.1002/nla.1858. |
[19] |
C. Li, Z. Chen and Y. Li,
A new eigenvalue inclusion set for tensors and its applications, Linear Algebra Appl., 481 (2015), 36-53.
doi: 10.1016/j.laa.2015.04.023. |
[20] |
C. Li, J. Zhou and Y. Li,
A new Brauer-type eigenvalue localization set for tensors, Linear Multiliear Algebra, 64 (2016), 727-736.
doi: 10.1080/03081087.2015.1119779. |
[21] |
C. Li, A. Jiao and Y. Li,
An $S $-type eigenvalue location set for tensors, Linear Algebra Appl., 493 (2016), 469-483.
doi: 10.1016/j.laa.2015.12.018. |
[22] |
L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing, 2005,129–132. Google Scholar |
[23] |
Q. Liu and Y. Li,
Bounds for the $ Z$-eigenpair of general nonnegative tensors, Open Math., 14 (2016), 181-194.
doi: 10.1515/math-2016-0017. |
[24] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
[25] |
L. Qi, G. Yu and E. X. Wu,
Higher order positive semidefinite diffusion tensor imaging, SIAM J. Imaging Sciences, 3 (2010), 416-433.
doi: 10.1137/090755138. |
[26] |
L. Qi,
Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput., 41 (2006), 1309-1327.
doi: 10.1016/j.jsc.2006.02.011. |
[27] |
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. |
[28] |
C. Sang, A new Brauer-type $ Z$-eigenvalue inclusion set for tensors, Numer. Algor., (2018), 1–14.
doi: 10.1007/s11075-018-0506-2. |
[29] |
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. |
[30] |
Y. Wang and G. Wang, Two $ S$-type $ Z$-eigenvalue inclusion sets for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 152, 12 pp.
doi: 10.1186/s13660-017-1428-6. |
[31] |
G. Wang, G. L. 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. |
[32] |
Y. Wang and L. Qi,
On the successive supersymmetric rank-1 decomposition of higher-order supersymmetric tensors, Numer. Linear Algebra Appl., 14 (2007), 503-519.
doi: 10.1002/nla.537. |
[33] |
T. Zhang and G. H. Golub,
Rank-one approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.
doi: 10.1137/S0895479899352045. |
[34] |
J. Zhao, A new $ Z$-eigenvalue localization set for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 85, 9 pp.
doi: 10.1186/s13660-017-1363-6. |
[35] |
J. Zhao and C. Sang,
Two new eigenvalue localization sets for tensors and theirs applications, Open Math., 15 (2017), 1267-1276.
doi: 10.1515/math-2017-0106. |
show all references
References:
[1] |
A. Ammar, F. Chinesta and A. Falcó,
On the convergence of a greedy rank-one update algorithm for a class of linear systems, Arch. Comput. Methods Eng., 17 (2010), 473-486.
doi: 10.1007/s11831-010-9048-z. |
[2] |
B. D. Anderson, N. K. Bose and E. I. Jury,
Output feedback stabilization and related problems-solutions via decision methods, IEEE Trans. Automat. Control, AC20 (1975), 53-66.
doi: 10.1109/tac.1975.1100846. |
[3] |
L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, Med. Image Comput. Comput. Assist. Interv., 5241 (2008), 1–8. Available from: https://www.ncbi.nlm.nih.gov/pubmed/18979725.
doi: 10.1007/978-3-540-85988-8_1. |
[4] |
N. K. Bose and P. S. Kamt,
Algorithm for stability test of multidimensional filters, IEEE Trans. Acoust. Speech Signal Process, 22 (1974), 307-314.
doi: 10.1109/TASSP.1974.1162592. |
[5] |
N. K. Bose and R. W. Newcomb,
Tellegon's theorem and multivariate realizability theory, Int. J. Electron, 36 (1974), 417-425.
doi: 10.1080/00207217408900421. |
[6] |
K. C. Chang, K. J. 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. |
[7] |
R. A. Devore and V. N. Temlyakov,
Some remarks on greedy algorithms, Adv. comput. Math., 5 (1996), 173-187.
doi: 10.1007/BF02124742. |
[8] |
A. Falco and A. Nouy,
A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach, J. Math. Anal. Appl., 376 (2011), 469-480.
doi: 10.1016/j.jmaa.2010.12.003. |
[9] |
J. He,
Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20 (2016), 1290-1301.
|
[10] |
J. He, Y. M. Liu, H. Ke, J. K. Tian and X. Li,
Bounds for the $Z$-spectral radius of nonnegative tensors, Springerplus, 5 (2016), 1727.
doi: 10.1186/s40064-016-3338-3. |
[11] |
J. He and T. Z. 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. |
[12] |
J. C. Hsu and A. U. Meyer, Modern Control Principles and Applications, The McGraw-Hill Series in Advanced Chemistry McGraw-Hill Book Co., Inc., New York-Toronto-London, 1956. |
[13] |
E. Kofidis and P. A. 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. |
[14] |
T. G. Kolda and J. R. Mayo,
Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095-1124.
doi: 10.1137/100801482. |
[15] |
L. D. Lathauwer, B. D. Moor and J. Vandewalle,
On the best rank-1 and rank-($R_1,R_2,\ldots ,R_N $) approximation of higer-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.
doi: 10.1137/S0895479898346995. |
[16] |
W. Li, D. Liu and S. W. Vong,
$Z$-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl., 483 (2015), 182-199.
doi: 10.1016/j.laa.2015.05.033. |
[17] |
C. Li and Y. Li,
An eigenvalue localization set for tensors with applications to determine the positive (semi-)definitenss of tensors, Linear Multilinear Algebra, 64 (2016), 587-601.
doi: 10.1080/03081087.2015.1049582. |
[18] |
C. Li, Y. Li and X. Kong,
New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 21 (2014), 39-50.
doi: 10.1002/nla.1858. |
[19] |
C. Li, Z. Chen and Y. Li,
A new eigenvalue inclusion set for tensors and its applications, Linear Algebra Appl., 481 (2015), 36-53.
doi: 10.1016/j.laa.2015.04.023. |
[20] |
C. Li, J. Zhou and Y. Li,
A new Brauer-type eigenvalue localization set for tensors, Linear Multiliear Algebra, 64 (2016), 727-736.
doi: 10.1080/03081087.2015.1119779. |
[21] |
C. Li, A. Jiao and Y. Li,
An $S $-type eigenvalue location set for tensors, Linear Algebra Appl., 493 (2016), 469-483.
doi: 10.1016/j.laa.2015.12.018. |
[22] |
L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing, 2005,129–132. Google Scholar |
[23] |
Q. Liu and Y. Li,
Bounds for the $ Z$-eigenpair of general nonnegative tensors, Open Math., 14 (2016), 181-194.
doi: 10.1515/math-2016-0017. |
[24] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
[25] |
L. Qi, G. Yu and E. X. Wu,
Higher order positive semidefinite diffusion tensor imaging, SIAM J. Imaging Sciences, 3 (2010), 416-433.
doi: 10.1137/090755138. |
[26] |
L. Qi,
Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput., 41 (2006), 1309-1327.
doi: 10.1016/j.jsc.2006.02.011. |
[27] |
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. |
[28] |
C. Sang, A new Brauer-type $ Z$-eigenvalue inclusion set for tensors, Numer. Algor., (2018), 1–14.
doi: 10.1007/s11075-018-0506-2. |
[29] |
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. |
[30] |
Y. Wang and G. Wang, Two $ S$-type $ Z$-eigenvalue inclusion sets for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 152, 12 pp.
doi: 10.1186/s13660-017-1428-6. |
[31] |
G. Wang, G. L. 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. |
[32] |
Y. Wang and L. Qi,
On the successive supersymmetric rank-1 decomposition of higher-order supersymmetric tensors, Numer. Linear Algebra Appl., 14 (2007), 503-519.
doi: 10.1002/nla.537. |
[33] |
T. Zhang and G. H. Golub,
Rank-one approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.
doi: 10.1137/S0895479899352045. |
[34] |
J. Zhao, A new $ Z$-eigenvalue localization set for tensors, J. Inequal. Appl., 2017 (2017), Paper No. 85, 9 pp.
doi: 10.1186/s13660-017-1363-6. |
[35] |
J. Zhao and C. Sang,
Two new eigenvalue localization sets for tensors and theirs applications, Open Math., 15 (2017), 1267-1276.
doi: 10.1515/math-2017-0106. |

Method | |
Theorem 3.1, i.e., Corollary 4.5 of [29] | 23.0000 |
Theorem 3.3 of [16] | 22.8625 |
Theorem 3.4 of [35], where |
22.8521 |
Theorem 3.2, i.e., Theorem 4.5 of [31] | 22.8149 |
Theorem 4.7 of [31] | 22.7759 |
Theorem 2.9 of [23] | 22.7217 |
Theorem 3.5 of [9] | 22.7163 |
Theorem 4.6 of [31] | 22.6478 |
Theorem 6 of [10] | 22.6290 |
Theorem 3.3, i.e., Theorem 5 of [34] | 22.5000 |
Theorem 4 of [30], where |
22.4195 |
Theorem 3.4, i.e., Theorem 7 of [28] | 22.2122 |
Theorem 3.5 | 21.2604 |
Method | |
Theorem 3.1, i.e., Corollary 4.5 of [29] | 23.0000 |
Theorem 3.3 of [16] | 22.8625 |
Theorem 3.4 of [35], where |
22.8521 |
Theorem 3.2, i.e., Theorem 4.5 of [31] | 22.8149 |
Theorem 4.7 of [31] | 22.7759 |
Theorem 2.9 of [23] | 22.7217 |
Theorem 3.5 of [9] | 22.7163 |
Theorem 4.6 of [31] | 22.6478 |
Theorem 6 of [10] | 22.6290 |
Theorem 3.3, i.e., Theorem 5 of [34] | 22.5000 |
Theorem 4 of [30], where |
22.4195 |
Theorem 3.4, i.e., Theorem 7 of [28] | 22.2122 |
Theorem 3.5 | 21.2604 |
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