|
|
H-张量的新判定准则及其应用
|
Abstract:
H-张量在科学与工程实践等领域中有着重要的应用，但在实际中要判定H-张量是比较困难的。本文通过构造不同的正对角阵，结合不等式的放缩技巧，给出了H-张量比较实用的新判别条件。作为应用，给出了判定偶次齐次多项式正定性的新方法，并给出相应的数值算例，表明了新结论的有效性。
H-tensors have important 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, by constructing different pos-itive diagonal matrices and combining the technique of inequality reduction, new practical condi-tions for H-tensors are given. As applications, new methods for determining the positive definite-ness of even homogeneous polynomials are presented, and the validity of new results is verified by some numerical examples.
| [1] | Chang, K.C., Pearson, K. and Zhang, T. (2008) Perron-Frobenius Theorem for Nonnegative Tensors. Communications in Mathematical Sciences, 6, 507-520. https://doi.org/10.4310/CMS.2008.v6.n2.a12 |
| [2] | 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.
https://doi.org/10.1016/j.cam.2010.06.002 |
| [3] | Qi, L. (2005) Eigenvalues of a Real Supersymetric Tensor. Journal of Symbolic Computation, 40, 1302-1324.
https://doi.org/10.1016/j.jsc.2005.05.007 |
| [4] | Lim, L.H. (2005) Singular Values and Eigenvalues of Tensors: A Variational Approach. Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, Puerto Vallarta, Vol. 1, 129-132. |
| [5] | 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. https://doi.org/10.1137/100813671 |
| [6] | 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. https://doi.org/10.1016/j.cam.2013.04.022 |
| [7] | Kannana, M.R., Mondererb, N.S. and Bermana, A. (2015) Some Properties of Strong H-Tensors and General H-Tensors. Linear Algebra and Its Applications, 476, 42-55. https://doi.org/10.1016/j.laa.2015.02.034 |
| [8] | Ding, W., Qi, L. and Wei, Y. (2013) M-Tensors and Nonsingular M-Tensors. Linear Algebra and Its Applications, 439, 3264-3278. https://doi.org/10.1016/j.laa.2013.08.038 |
| [9] | Wang, F. and Sun, D.S. (2016) New Criteria for H-Tensors and an Application. Journal of Inequalities and Applications, 96, 1-12. https://doi.org/10.1186/s13660-016-1041-0 |
| [10] | Wang, F., Sun, D.S., Zhao, J.X., et al. (2017) New Practical Cri-teria for H-Tensors and Its Application. Linear and Multilinear Algebra, 65, 269-283. https://doi.org/10.1080/03081087.2016.1183558 |
| [11] | Li, Y.T., Liu, Q.L. and Qi, L. (2017) Programmable Criteria for Strong H-Tensors. Numerical Algorithms, 74, 199-221.
https://doi.org/10.1007/s11075-016-0145-4 |
| [12] | Zhang, L.P., Qi, L. and Zhou, G.L. (2014) M-Tensors and Some Applications. SIAM Journal on Matrix Analysis and Applications, 35, 437-542. https://doi.org/10.1137/130915339 |
| [13] | Zhang, K.L. and Wang, Y.J. (2016) An H-Tensor Based Iterative Scheme for Identifying the Positive Definiteness of Multivariate Homogeneous Forms. Journal of Computational and Applied Mathematics, 305, 1-10.
https://doi.org/10.1016/j.cam.2016.03.025 |
| [14] | Bai, D.J. and Wang, F. (2021) New Methods Based H-Tensors for Identifying the Positive Definiteness of Multivariate Homogeneous Forms. AIMS Mathematics, 6, 10281-10295. https://doi.org/10.3934/math.2021595 |
| [15] | Qi, L. and Song, Y.S. (2014) An Even Order Symmetric B-Tensor Is Positive Definite. Linear Algebra and Its Applications, 457, 303-312. https://doi.org/10.1016/j.laa.2014.05.026 |
| [16] | Li, C.Q. and Li, Y.T. (2015) Double B-Tensors and Quasi-Double B-Tensors. Linear Algebra and Its Applications, 466, 343-356. https://doi.org/10.1016/j.laa.2014.10.027 |
| [17] | Bai, D.J. and Wang, F. (2022) New Criterions-Based H-Tensors for Testing the Positive Definiteness of Multivariate Homo-geneous Forms. Mathematics, 10, Article No. 2416. https://doi.org/10.3390/math10142416 |
| [18] | Lv, C.Q. and Ma, C.F. (2021) An Iterative Scheme for Identifying the Positive Semi-Definiteness of Even-Order Real Symmetric H-Tensor. Journal of Computational and Applied Mathematics, 392, Article ID: 113498.
https://doi.org/10.1016/j.cam.2021.113498 |
