New methods based $ \mathcal{H} $-tensors for identifying the positive definiteness of multivariate homogeneous forms

Abstract

<abstract><p>Positive definite polynomials are important in the field of optimization. $ \mathcal{H} $-tensors play an important role in identifying the positive definiteness of an even-order homogeneous multivariate form. In this paper, we propose some new criterion for identifying $ \mathcal{H} $-tensor. As applications, we give new conditions for identifying positive definiteness of the even-order homogeneous multivariate form. At last, some numerical examples are provided to illustrate the efficiency and validity of new methods.</p></abstract>

New Criterions-Based H-Tensors for Testing the Positive Definiteness of Multivariate Homogeneous Forms

“…In this paper, we continue to propose some new sufficient conditions of H-tensors, and we present a new non-parameter-involved iterative algorithm for testing H-tensors. The obtained results extend the corresponding results in [17][18][19][20][21]. The validity of new methods is theoretically guaranteed, and the numerical experiments show their efficiency.…”

“…Subsequently, based on the notion of diagonally dominant tensors, various criterions for H-tensors are established [13][14][15][16][17][18][19][20][21], which only depend on the elements of the tensors, and are very effective for testing whether a tensor is an H-tensor or not. For example,…”

“…Therefore, A does not satisfy the conditions of Theorem 1 (Theorem 1 in [17]), Theorem 2 (Theorem 1 in [18]), and Theorem 3 (Theorem 2 in [21]), respectively. Remark 1.…”