Split Iteration And CG-like Algorithm For Solving Several Kinds Of Constraint Matrix Equations

The problems of solving linear matrix equations subject to constraints and their approximation have wide applications in fields such as control theory,vibration theory,model correction,image restoration,and civil engineering.Therefore,the problem of solving linear matrix equations is of great research significance.This doctoral thesis investigates some split iteration and CG-like algorithms for solving several types of constrained matrix equations.The main work are as follows:Firstly,for the constrained matrix equation AXB=C,where A is a nonsymmetric positive definite matrix and B is a symmetric positive definite matrix,a new iterative method is proposed based on the PHSS split iteration idea and combined with the G1-CG algorithm and Gl-FOM algorithm to solve the general solution.The convergence of the new algorithm is proved.And numerical experiments show that this iterative method is feasible and effective.Secondly,for the matrix equation AXB=C,where A is a non-symmetric positive definite matrix and B is a symmetric positive definite matrix,the problem of finding a solution X with symmetric structure is considered.A new numerical method is proposed and the sufficient conditions for convergence and convergence proof for the numerical iteration method are presented.Numerical experiments show that this method is effective.Thirdly,for the matrix equation ∑i=1t AiXiBi=E with submatrix constraints,this work discusses the problem of finding a mixed solution matrix group[X1,X2,…,Xt]and its best approximation solution.A new CG-like algorithm(i.e.gradient-based algorithm)is proposed for this problem.The convergence of the iterative algorithm is proved,and a numerical solution method for the optimal approximation problem related to this type of problem is presented.This algorithm is applied to image restoration,and numerical experiments and applications demonstrate its effectiveness and feasibility.Finally,for the linear matrix equation AXB=C,where A and B are both non-symmetric positive definite matrices,the problem of finding a solution X with symmetric structure is considered.This work combines HSS and PHSS split iteration with the CG-like algorithm proposed in the third research to propose two new algorithms for finding a symmetric solution or a non-symmetric solution to AXB=C(where A and B are both non-symmetric positive definite matrices).The convergence conditions and convergence proof for this algorithm are presented.Numerical experiments show that this method is not only effective but also feasible.