Iterative Methods For Solving Some Special Tensor Problems

The tensor has a wide range of practical applications in different subject areas and deeper and deeper research.It has been a hot topic.Many scholars devoted themselves to study some popular directions like tensor decomposition,iterative methods of tensor equations,tensor complementary problems and so on.In this paper,we do some theoretical analysis and algorithm improvements based on existing research results for several problems related to tensors.The main content of this paper is organized as follows:In Chapter 1,we mainly introduce application background and research status of the tensor.Besides,we introduce some basic definitions of a tensor and the algorithm that we need in this paper.In Chapter 2,we mainly study using a modified symmetric orthogonal decomposition method to approximate a symmetric tensor.To discuss the symmetric orthogonal approximation of a symmetric tensor,we firstly translate it into a equation constrained problem for theoretical analysis.In our algorithm,we use an adaptive shifted power method to compute generalized tensor eigenvectors and propose the convergence analysis of this algorithm.In the end,this algorithm is verified effectiveness through numerical experiments.In Chapter 3,the biconjugate gradient stabilized(BCGSTAB)method is a fast and smoothly converging variant of biconjugate gradient(Bi CG)method.We generalize BCGSTAB method to solve the Stein tensor equation.We raise the algorithm of tensor format and specific proof process of existence of a solution.And we present the convergence theorem of this algorithm.Numerical experiments demonstrate this algorithm is effective and feasible for solving the Stein tensor equation.In Chapter 4,based on the existing tensor splitting method for solving multilinear equations and the structure of tensor absolute value equation,the existence of the solution of absolute value equation is proved.And we propose the tensor splitting method to solve this equation.In addition,convergence analysis is also proved under appropriate conditions.Finally,numerical experiments show that the proposed algorithm is an effective iterative algorithm.In Chapter 5,we sum up the research content of this paper and put forward the future research direction.