New iterative criteria for strong H $\mathcal{H}$ -tensors and an application

Abstract

Strong -tensors play an important role in identifying the positive definiteness of even-order real symmetric tensors. In this paper, some new iterative criteria for identifying strong -tensors are obtained. These criteria only depend on the elements of the tensors, and it can be more effective to determine whether a given tensor is a strong -tensor or not by increasing the number of iterations. Some numerical results show the feasibility and effectiveness of the algorithm.

Some Criteria for $${\varvec{\mathcal {H}}}$$-Tensors

“…In this paper, we establish some new criteria in identifying nonsingular H-tensors and give two numerical examples. These criteria are different from those of [1,6,7,9,10,12].…”

“…In [6,7], the authors provided an iterative algorithm for strong H-tensors, respectively. In [1,9,10,12], the authors gave some criteria for H-tensors. In [8], the authors gave some properties and conditions for H-tensors and nonsingular H-tensors.…”

“…and 0 ⩽ ⩽ 1 2 , such that and there is i 0 ∈ J , such that (7) holds strictly, then is a nonsingular H-tensor.…”

“…The H-tensor which contains the M-tensor as special cases plays an important role in identifying positive definiteness of even-order real symmetric tensors [2,3,5,11]. In [1,6,7,9,10,12], the authors presented some criteria for H-tensors. In [8], the authors gave some properties and conditions for H-tensors and nonsingular H-tensors.…”

“…For an m-th order n-dimensional tensor = (a i 1 i 2 ⋯i m ) with nonzero diagonal entries, if there exists a partition (J, J) of the index set [n] and 0 ⩽ ⩽1 2 , such that then is a nonsingular H-tensor.…”