A Study Of Algorithms For A Class Of Matrix Equation And Tensor Equation

The main work of this dissertation involve two parts.The first part focuses on the numerical method for large linear matrix equation,We get an iterative algorithm based on the ideal of rational Krylov subspace methods,Galerkin projection and Kronecker product structure of linear systems.The second part aims to study the numerical methods for linear systems from the aspect of tensor equation.By using the hierarchical identification principle and tensor calculus,an iterative algorithm based on gradient is established to solve a class of special tensor equation and then we extend the iterative to solve more general tensor equations.The detailed content of this thesis are as follows:Chapter one and chapter two introduce the background of the related issues of matrix and tensor.We roughly sort out their history and current development status,and then introduce various operations and properties of matrix and tensor which this paper involves.In chapter three,we use the truncated low-rank methods of [11] and the Krylov subspace methods of [9] putting forward a algorithm based on rational Krylov subspace methods and Galerkin projection for the numerical solutions of general matrix equations,and then we verify the validity of the algorithm by numerical experiments.Next,by using the hierarchical identification principle of [1,13] and tensorNext,by using the hierarchical identification principle of [1,13] and tensor calculus of [25] we get an iterative algorithm based on gradient for tensor equations (?) and(?),after that,the algorithm's validity is proved by numerical experiments and extend the algorithm to solve more general tensor equation(?) and then we verify the feasibility of the algorithm by numerical experiments.In addition,we simply discuss the tensor equation (?) but don't explore it's related properties.we briefly...