<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>
Positive definite homogeneous multivariate forms play an important role in polynomial problems and medical imaging, and the definiteness of forms can be tested using structured tensors. In this paper, we state the equivalence between the positive definite multivariate forms and the corresponding tensors, and explain the connection between the positive definite tensors with H-tensors. Then, based on the notion of diagonally dominant tensors, some criteria for H-tensors are presented. Meanwhile, with these links, we provide an iterative algorithm to test the positive definiteness of multivariate homogeneous forms and prove its validity theoretically. The advantages of the obtained results are illustrated by some numerical examples.
Positive definite polynomials are important in the field of optimization. ℋ-tensors play an important role in identifing the positive definiteness of an even-order homogeneous multivariate form. In this paper, we propose an iterative scheme for identifying ℋ-tensor and prove that the algorithm can terminate within finite iterative steps. Some numerical examples are provided to illustrate the efficiency and validity of methods.
A positive definite homogeneous multivariate form plays an important role in the field of optimization, and positive definiteness of the form can be identified by a special structured tensor. In this paper, based on the equivalence between the form and the corresponding tensor, and the links of the positive definiteness of a tensor with ℋ-tensor, we propose an ℋ-tensor-based criterion for identifying the positive definiteness of multivariate homogeneous forms. Some numerical examples are provided to illustrate the efficiency and validity of our results.
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