New iterative criterions for testing the positive definiteness of multivariate homogeneous forms

The positive definite homogeneous multivariate forms play an important role in the automatic control and medical imaging, and the definiteness of the forms can be identified by special structured tensors. In this paper, we first state the equivalence between the positive definite multivariate forms and the corresponding tensors and account for the links between the positive definite tensors with H-tensors. Then based on diagonal dominance, some iterative criterions are presented to test H-tensors. Furthermore, we establish new iterative schemes for testing the positive definite multivariate homogeneous forms. The efficiency and validity of new methods are illustrated by numerical examples.