An HH-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms

Identifying the positive definiteness of an even-order homogeneous multivariate form is an important task due to its wide applications in such as medical imaging and the stability analysis of nonlinear autonomous systems via Lyapunov’s direct method in automatic control and multivariate network realizability analysis. In this paper, based on the equivalence of the positive definiteness of the form to that of the underlying tensor, and the links between the positive definiteness of a tensor with strong HH-tensor, we propose an HH-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms. The validity of the iterative scheme is guaranteed theoretically and the given numerical experiments show the efficiency of the scheme.