Research On The Judgment Algorithm Of The Homogeneous Polynomial Positive Definiteness Based On Strong H- Tensor

Tensor as a newly developed tool in analysis of large data is an extension of the matrix.And -tensor as an extension of -matrix owns specially structure and plays a significant role in tensor analysis and computing.The structural properties,various criteria and iterative schemes for tensors and strong -tensors are explored in the literature.The tensor which is reducible or not directly affects the design and implementation model of the scheme.Thus,researches of the reducibility enjoys great popularity.Moreover,the identification of 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.And the positive definiteness of polynomial with its corresponding tensor is consistent and strong -tensor is equivalent with the positive definiteness of tensor,so we can not help but to take advantage of the relationship between them,solve the problem of the positive definiteness of the polynomial based on tensor.In this paper,we explore the properties of -tensors and establish some new characterizations of strong -tensors via weak reducibility and the principal subtensor.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 -tensor,we propose an -tensor based iterative scheme for identifying the positive definiteness of a multivariate homogeneous form.We present two non-parameter implementation iterative schemes for identifying strong -tensors.The validity of the scheme is offered.The numerical experiments are also provided to show the efficiency of the scheme.The structure of this paper is organized as follows:In chapter 1,the main content is to present the research background and development status of the tensor and the positive definiteness of a multivariate homogeneous form,and the principal results of this paper.In chapter 2,the core content is given the definition and basic properties of the strong-tensor.Then establish some new characterizations for strong -tensors via weak reducibility and the principal subtensor.And the results obtained extend the corresponding conclusions for strong -matrices and improve the existing results for strong -tensors.In chapter 3,the principal content is built on the equivalence of the positive definiteness of the form to that of the underlying tensor.By means of the links between the positive definiteness of a tensor and strong -tensor,we present two non-parameter implementation iterative schemes based on -tensor for identifying the positive definiteness of a multivariate homogeneous form.The validity of the schemes is provided.The numerical experiments are also given to show the efficiency of the scheme.In chapter 4,the main content is to provide a brief summary of the content,and on the outlook for future work to be done.