H-张量的新判定及其应用
New Criteria for H-Tensors and Its Application
DOI: 10.12677/AAM.2020.95088, PDF,    科研立项经费支持
作者: 柏冬健, 徐玉梅, 吴 念：贵州民族大学数据科学与信息工程学院，贵州 贵阳
关键词: H-张量；实对称张量；不可约；非零元素链；正定性；H-Tensors； Real Symmetric Tensors； Irreducible； Nonzero Elements Chain； Positive Definiteness
摘要: H-张量在科学和工程实际中具有重要应用，但在实际中要判定H-张量是比较困难的。通过构造不同的正对角阵，结合不等式的放缩技巧，给出了一些比较实用的新判别条件。作为应用，给出了判定偶数阶实对称张量正定性的条件，相应数值算例说明了新结果的有效性。
Abstract: H-tensors have wide applications in science and engineering, but it is difficult to determine whether a given tensor is an H-tensor or not in practice. In this paper, we give some practical conditions for H-tensors by constructing different positive diagonal matrices and applying some techniques of inequalities. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor are given. Advantages of results obtained are illustrated by numerical examples.
文章引用：柏冬健, 徐玉梅, 吴念. H-张量的新判定及其应用[J]. 应用数学进展, 2020, 9(5): 742-751. https://doi.org/10.12677/AAM.2020.95088

参考文献

[1] Chang, K.C., Pearson, K. and Zhang, T. (2008) Perron-Frobenius Theorem for Nonnegative Tensors. Communications in Mathematical Sciences, 6, 507-520. [Google Scholar] [CrossRef] 
[2] Lathauwer, L.D., Moor, B.D. and Vandewalle, J. (2000) On the Best Rank-1 and Rank-(R1,R2,RN) Approximation of Higher-Order Tensors. SIAM Journal on Matrix Analysis and Applications, 21, 1324-1342. [Google Scholar] [CrossRef] 
[3] Liu, Y.J., Zhou, G.L. and Ibrahim, N.F. (2010) An Always Convergent Algorithm for the Largest Eigenvalue of an Irreducible Nonnegative Tensor. Journal of Computational and Applied Mathematics, 235, 286-292. [Google Scholar] [CrossRef] 
[4] Ng, M., Qi, L.Q. and Zhou, G.L. (2010) Finding the Largest Eigenvalue of a Nonnegative Tensor. SIAM Journal on Matrix Analysis and Applications, 31, 1090-1099. [Google Scholar] [CrossRef] 
[5] Zhang, T. and Golub, G.H. (2001) Rank-One Approximation to Higher-Order Tensors. SIAM Journal on Matrix Analysis and Applications, 23, 534-550. [Google Scholar] [CrossRef] 
[6] Qi L. (2005) Eigenvalues of a Real Supersymetric Tensor. Journal of Symbolic Computation, 40, 1302-1324. [Google Scholar] [CrossRef] 
[7] Kofidis, E. and Regalia, P.A. (2002) On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors. SIAM Journal on Matrix Analysis and Applications, 23, 863-884. [Google Scholar] [CrossRef] 
[8] Yang, Y.N. and Yang, Q.Z. (2011) Further Results for Perron-Frobenius Theorem for Nonnegative Tensors II. SIAM Journal on Matrix Analysis and Applications, 32,1236-1250. [Google Scholar] [CrossRef] 
[9] Ni, Q., Qi, L. and Wang, F. (2008) An Eigenvalue Method for Testing Positive Definiteness of a Multivariate Form. IEEE Transactions on Automatic Control, 53, 1096-1107. [Google Scholar] [CrossRef] 
[10] Zhang, L.P., Qi, L.Q. and Zhou, G.L. (2014) M-Tensors and Some Applications. SIAM Journal on Matrix Analysis and Applications, 35, 437-542. [Google Scholar] [CrossRef] 
[11] Kannana, M.R., Mondererb, N.S. and Bermana, A. (2015) Some Properties of Strong H-Tensors and General H-Tensors. Linear Algebra & Its Applications, 476, 42-55. [Google Scholar] [CrossRef] 
[12] Ding, W., Qi, L.Q. and Wei, Y.M. (2013) M-tensors and Nonsingular M-Tensors. Linear Algebra and Its Applications, 439, 3264-3278. [Google Scholar] [CrossRef] 
[13] Li, C.Q., Wang, F., Zhao, J.X., et al. (2014) Criterions for the Positive Definiteness of Real Supersymmetric Tensors. Journal of Computational and Applied Mathematics, 255, 1-14. [Google Scholar] [CrossRef] 
[14] , F. and Sun, D.S. (2016) New Criteria for H-Tensors and an Application. Journal of Inequalities and Applications, 96, 1-12. [Google Scholar] [CrossRef] 
[15] Li, Y.T., Liu, Q.L. and Qi, L.Q. (2017) Programmable Criteria for Strong H-Tensors. Numerical Algorithms, 74, 199-221. [Google Scholar] [CrossRef] 
[16] Qi, L.Q. and Song, Y.S. (2014) An Even Order Symmetric B-Tensor Is Positive Definite. Linear Algebra and Its Applications, 457, 303-312. [Google Scholar] [CrossRef] 
[17] Li, C.Q. and Li, Y.T. (2015) Double B-Tensors and Quasi-Double B-Tensors. Linear Algebra and Its Applications, 466, 343-356. [Google Scholar] [CrossRef] 

为你推荐