Study On Numerical Algorithm For Several Classes Of Tensor Equations

Tensors play an important role in describing large-scale high-dimensional problems,such as communication records,quantum science,multi-linear image analysis for face recognition,etc.In view of the wide application of tensors,tensor equations have become a hot topic in the field of computational mathematics,so many scholars are devoted to how to solve tensor equations quickly and effectively.Based on the existing research results,several different types of tensor equations are discussed in detail in this paper.In chapter 1,we introduce the application background,research significance and present situation of tensor equation.In addition,some basic concepts of tensors are introduced.In chapter 2,an accelerated LM algorithm for solving tensor absolute value equation is obtained by improving the classical Levenberg-Marquardt(LM)method.Theoretical analysis shows that the iterative sequences generated by the algorithm converge under proper conditions.Numerical experiments show that the algorithm is feasible.In chapter 3,combined with classical LM method and its variant,we study the tensor equation with Hankel structure,and propose a correction adaptive LM algorithm for solving Hankel tensor equation.It is proved that this method has second convergence.In chapter 4,the modified conjugate gradient(MCG)method and the conjugate gradient least squares(CGLS)method are proposed to solve the generalized complex Sylvester tensor equations.They are all based on the classical Krylov subspace method for solving linear equations.Theoretical analysis show that the proposed algorithms converge to Sylvester tensor equation solutions for any initial tensor,excluding rounding errors.The numerical results show that the MCG algorithm and CGLS algorithm are effective.In Chapter 5,the thesis summarizes the research work and points out the following research ideas and problems to be solved.