# Some classes of interval tensors and their properties

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Sciences, University of Hormozgan, P. O. Box 3995, Bandar Abbas, Iran

10.22072/wala.2022.550031.1369

Abstract

First, we define and investigate some new classes of interval tensors, called interval exceptionally regular tensors  ($ER-$tensor) and interval $wP-$tensors which is relevant to interval strictly semi-positive tensors. Also, we show that $ER-$tensor is a wide class of interval tensors, which includes many important structured tensors. Second, some classes of interval matrices are extended to interval tensors, such as interval $R(R_0)$-tensor and column sufficient interval tensor. We discuss their relationships with interval positive semi-definite tensors and some other structured interval tensors. In addition,  necessary and sufficient conditions for interval (strictly) copositive and interval $E_0$-tensors are presented and investigated. Finally, we extend the concept of the column sufficient interval matrix to the column sufficient interval tensor.

Keywords

#### References

[1] H. Bozorgmanesh, M. Hajarian and A.Th. Chronopoulos, Interval Tensors and their application in solving multi-linear
systems of equations, Comput. Math. Appl., 79(3) (2020), 697-715.
[2] H. Chen, L. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Front. Math. China, 13(2)
(2018), 255-276.
[3] R.W. Cottle, J.-Sh. Pang and R.E. Stone, The Linear Complementarity Problem, Society for Industrial and Applied
Mathematics, 2009.
[4] R.W. Cottle, J.-Sh. Pang and V. Venkateswaran, Sufficient matrices and the linear complementarity problem, Linear
Algebra Appl., 114 (1989), 231-249.
[5] W. Ding, Z. Luo and L. Qi., P-Tensors, P_0-Tensors, and Tensor Complementarity Problem,  arXiv preprint:
arXiv:1507.06731, 2015.
[6] W. Ding, L. Qi and Y. Wei, $\mathcal{M}$-tensors and nonsingular $\mathcal{M}$-tensors, Linear Algebra Appl.,
439(10) (2013), 3264-3278.
[7] M. Fiedler, J. Nedoma, J. Ramík, J. Rohn and K. Zimmermann, Linear Optimization Problems with Inexact Data, Springer,
New York, 2006.
[8] M. Heyouni, F. Saberi-Movahed and A. Tajaddini, A tensor format for the generalized Hessenberg method for solving
Sylvester tensor equations, J. Comput. Appl. Math., 377 (2020),112878.
[9] M. Hladík, Stability of the linear complementarity problem properties under interval uncertainty, CEJOR, Cent. Eur. J.
Oper. Res., 29(3) (2021), 875-889.
[10] M. Hladík, D. Daney and E. Tsigaridas, Characterizing and approximating eigenvalue sets of symmetric interval
matrices, Comput. Math. Appl., 62(8) (2011), 3152-3163.
[11] T.G. Kolda and W.B. Brett, Tensor decompositions and applications, SIAM Rev., 51(3) (2009), 455-500.
[12] R.E. Moore, Interval Arithmetic and Automatic Error Analysis in Digital Computing (Ph.D. thesis), Stanford Univ Calif
Applied Mathematics And Statistics Labs, 1962.
[13] R.E. Moore, R.B. Kearfott and M.J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia, PA, 2009.
[14] S. Rahmati and M.A. Tawhid, On intervals and sets of hypermatrices (tensors), Front. Math. China, 15(6) (2020),
1175-1188.
[15] J. Rohn, Bounds on eigenvalues of interval matrices, ZAMM, Z. Angew. Math. Mech., 78(3) (1998), 24-27.
[16] S.M. Rump., Fast interval matrix multiplication, Numer. Algorithms, 61(1) (2012), 1-34.
[17] Y. Song and L. Qi, Properties of some classes of structured tensors, J. Optim. Theory Appl., 165(3) (2015), 854-873.
[18] Y. Song and L. Qi, Properties of tensor complementarity problem and some classes of structured tensors,
arXiv:1412.0113v1, 2014.
[19] Y. Wang, Zh.-H. Huang and X.-L. Bai, Exceptionally Regular Tensors and Tensor Complementarity Problems,
Optimization Methods and Software, 31(4) (2016), 815-828.
[20] Y. Yang and Q. Yang, Further results for Perron–Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal.
Appl., 31(5) (2010), 2517-2530.
[21] L. Zhang, L. Qi and G. Zhou, M-tensors and some applications, SIAM J. Matrix Anal. Appl., 35(2) (2014), 437-452.