Research On Tensor Positive Definite Decision Algorithm Based On H - Tensor

Tensors have wide applications in numerous application domains recently, and there come after lots of problems. For instance, to identify the positive definiteness of homogeneous multivariate polynomial, which plays an important role in the stability study of nonlinear autonomous systems via Lyapunov’s direct method in automatic control such as the multivariate network realizability theory, a test for Lyapunov stability in multivariate filters, a test of existence of periodic oscillations using Bendixon’s theorem, and the output feedback stabilization problems. It becomes very important to identify the positive definiteness of homogeneous multivariate polynomial and it has also received much attention of researchers in recent decade. The strong the links between the positive definiteness of a tensor with strong H-tensor has been found. Someone pointed out that homogeneous multivariate polynomial is positive definite if and only if the real supersymmetric underlying tensor is positive definite, which is a problem of identifying strong H-tensors. H-tensor is developed based on the comparison tensor and the ?-tensor, which plays an important role in science and engineering. Strong H-tensor plays an important role in science and engineering, but it is difficult to determine whether a given tensor is a strong H-tensor or not in practice.In this paper, based on the strong H-tensor’s typical feature of strictly generalized diagonal dominance and the equivalence of the positive definiteness of the form to that of the underlying tensor, we propose some iterative schemes to identify strong H-tensors, then to identify the positive definiteness of tensors, further to identify the positive definiteness of a multivariate homogeneous form. The validity of the iterative scheme is guaranteed theoretically and the numerical experiments are also given to show the efficiency of the scheme.