Researches On The Iterative Algorithms For Several Classes Of Matrix Equations And Tensor Equations

Many problems in scientific computations and engineering applications can be re-duced to solving various linear matrix equations.Especially,the discrete-time periodic matrix equations have been used as a main tool of analysis in cyclostationary and stochas-tic processes,Luenberger-type observers design problem for linear discrete-time periodic systems,signal processing,periodic robust state-feedback pole assignment problems and output-feedback optimal periodic control problems.The inverse quadratic eigenvalue prob-lem appears in the acoustic simulation of pro-elastic materials,the elastic deformation of anisotropic materials and finite element discretization in structural analysis.As the exten-sion of matrix computation,tensor computation is the latest research hotspot in the last twenty years.Various kinds of tensor equations arise from mechanics,physics,Markov pro-cess,control theory,partial differential equations and engineering problems.The radiation transfer equation,high-dimensional Possion equation,Einstein gravitational field equation and piezoelectric effect equation are all tensor equations.Especially,by means of the spec-trum allocation method,the discretization of the radiation transfer equation gives rise to a Sylvester tensor equation via the Tucker product.The discretization of the Possion problem gives rise to a tensor equation via the Einstein product.This thesis is concerned with the iterative methods for several kinds of matrix equations and tensor equations.The main achievements of this paper are as follows:In Chapter 1,we consider the least squares solutions of a class of linear periodic matrix equations and the corresponding optimal approximate problem.By means of the properties of projection,the KKT conditions of the minimization problem in the linear subspace and the periodicity of the equation,we derive the normal equations.Then,we propose a finite iterative algorithm to find the least squares solutions of the linear periodic matrix equations over symmetric periodic matrices and present the convergence analysis.By choosing a special kind of the initial matrices,the unique solution group with the least Frobenius norm can be obtained.Furthermore,in the solution set of the above problem,the unique optimal approximation solution group to a given matrix group in the Frobenius norm can be derived by finding the unique symmetric periodic least squares solution with the least Frobenius norm of a new corresponding minimum Frobenius norm problem.The numerical examples are provided to illustrate that the proposed algorithm can obtain the symmetric periodic least squares solution in a finite number of iterative steps.In Chapter 2,we propose the iterative methods for the bisymmetric least squares solu-tions of the inverse quadratic eigenvalue problem with a submatrix constraint and its corre-sponding optimal approximation problem.Unlike the construction process of the conjugate gradient method for many linear matrix equations,we establish the iterative algorithm by means of the nonlinear conjugate gradient method for the convex quadratic programming problem.And the global linear convergence of the proposed algorithm is proved by us-ing different methods.Furthermore,we establish the iterative algorithm for the optimal approximation problem.Numerical examples show that the proposed algorithm is effective.In Chapter 3,for the Sylvester tensor equation via the Tucker product,we first con-struct the tensor form of the global generalized Hessenberg process with pivoting in order to generate a linearly independent tensor basis for the tensor Krylov subspace,and then establish two Hessenberg-based methods:CMRH-BTF method and Hess-BTF method by imposing the minimal residual norm condition and the Petrov-Galerkin condition,respec-tively.Secondly,by reformulating the Sylvester tensor equation as a operator equation,we establish the tensor form of the global LSMR method by means of the operator bidiagonal-ization process,and then present the implementation details.Thirdly,we present the tensor form of the conjugate gradient least squares method and prove that the solution(or the least squares solution)of the Sylvester tensor equation can be obtained within a finite number of iterative steps in the absence of round-off errors.By selecting the appropriate initial tensor,the least Frobenius norm solution(or the least Frobenius norm least squares solution)of the tensor equation can be obtained.Finally,we show the efficiency and superiority of the proposed algorithms in this chapter by some numerical examples.In Chapter 4,for the tensor equation via the Einstein product A*N χ＝c,we first construct the tensor forms of the Arnoldi and Lanczos processes in order to form the or-thonormal tensor basis of the tensor Krylov subspace,and establish the tensor form of the global GMRES method,the tensor form of the MINIRES method(denoted by MINIRES-BTF)and the tensor form of the SYMMLQ method(denoted by SYMMLQ-BTF).We also describe the implementation details of the MINIRES-BTF and SYMMLQ-BTF methods.ative steps in the absence of round off errors.For the tensor equation A*Nχ*MB+C*NχD=F,we derive the tensor form of the CGLS method,the tensor form of the LSQR method and the tensor form of the LSMR method.Furthermore,numerical examples are provided to illustrate the superiority and efficiency of the proposed algorithms.In Chapter 5,we present an iterative algorithm to solve the tensor equation via the Einstein product A*NχMB=C with the tensor inequality constraint D*Nχ*Mε≥F.By using the polar decomposition of tensor,the Moore-Penrose generalized inverse of tensors and the Hilbert space decomposition theorem,we give the convergence analysis.Finally,numerical examples show the numerical performances of the proposed algorithm.