On Algorithms Of Constrained Matrix Equations And Their Optimal Approximation Problems

In this Ph.D.thesis,three problems are considered.First,matrix equation least squares solution under circulant matrices constraint conditions is discussed;Second,the matrix optimal approximation problem under two constraints is s-tudied by using two kinds of alternating projection algorithm;Third,the linear equations by using preconditioned iterative methods are solved.This paper is divided into five chapters,the main contents are as follows:In Chapter 1,we introduce the research background and research results of the problems under consideration:the constrained matrix equation,matrix opti-mal approximation problem and linear equations of the research background and research achievements,we give a brief statement and point out the research mo-tivation and significance of this thesis,we also summarize the main work of this thesis.In Chapter 2,we study the constrained least squares problem of matrix equa-tions AX = B,AXB = C and A1XB1= C1,A2XB2 =C2,obtain their expres-sions of general solution and the unique solution.Considering the constraints:X is generalized Toeplitz matrices,upper triangular Toeplitz matrices,lower trian-gular Toeplitz matrices,symmetric Toeplitz matrices,Hankel matrices,circulant matrices,skew circulant matrices,symmetric circulant matrices,skew symmetric circulant matrix,first-and-last-sum r-circulant matrices and first-and-last-sum r-retrocirculant matrices.The method used is different from the traditional direct method and iterative method,it is based on the special structure and properties of constraint matrices.In Chapter 3,we study the optimal approximation problem of the constraint matrix.There are two categories constraints,the first condition is matrix satisfy a consistent matrix equation or an inconsistent matrix equation(the constraints for the least squares solution).Another condition is the matrices are circulant matrices(chapter 2 situations).We consider the first-order matrix equations and we adopt the alternating projection algorithm.In Chapter 4,based on the linear equations Ax = b,there has been many research results on generalized two parameters overrelaxation algorithm(GTOR).In order to improve the convergence rate,we introduce five preconditioners,thus introduce the preconditioned generalized two parameters overrelaxation algorithm(PGTOR).On one hand,through theoretical derivation,we come to the conclusion that the preconditioned method has smaller convergence radius and faster iteration speed than the original one.On the other hand,we also make convergence analysis and comparison on several kinds of different PGTOR methods.The numerical results confirm the theoretical deduction.In Chapter 5,we further conclude the work done in this paper and look forward to the work we can continue to do.