Research On Two Kinds Of Tensor Equation Iterative Algorithms

Tensors are widely used in signal processing,image processing,non-linear optimiza-tion,higher-order statistics and data mining.Many problems in scientific and engineering calculations can be expressed in the form of tensor-vector products,called tensor equation.The tensor equation can be seen as a natural extension of the matrix equation=(7,which plays an important role in the scientific computing and engineering applications.How to effectively solve the tensor equation has very important theoretical significance and practical application value.Based on the existing research results,this paper conducts detailed theoretical analysis and algorithm research on the Hankel tensor equation and the strongtensor absolute value equation,and gives the proof of the convergence of these algorithms.Finally,the feasibility and effectiveness of these algorithms are verified through numerical experiments.The main content of this paper is organized as follows:In Chapter 1,the research background,research significance and research status of tensor equations at home and abroad are reviewed.In addition,some basic knowledge needed in this paper are also introduced.In Chapter 2,for the Hankel tensor equation,a fast Fourier transform is added to the L-M method,so that only the generated vector of the Hankel tensor is stored during the operation,instead of the complete tensor.The proof of the global convergence and the quadratic convergence under the local error bound of the algorithm are given.Finally,the fast Fourier L-M algorithm is applied to solve the(5-eigenpair of Hankel tensor and we verify the feasibility and effectiveness of the algorithm by numerical experiments.In Chapter 3,based on the unique structure of the Hankel tensor,we propose a modified BFGS algorithm(MBFGS)and prove that the algorithm has global convergence under non-convex condition.Finally,the feasibility and effectiveness of the algorithm are verified via numerical experiments.In Chapter 4,for the tensor absolute value equation(TAVEs),whose coefficient tensor A?₈,9))is a strongsymmetric tensor and>(B)+1,an algorithm which combines the non-monotone Armijo line search technology with the L-M algorithm is proposed,and the proof of the global convergence and the quadratic convergence under the local error bound are given.The numerical results further show that the proposed algorithm is effective.In Chapter 5,we summarize the research work of this paper,and put forward some problems that have yet to be solved and the ideas for future research work.