Research On Iterative Algorithms For Solving Several Types Of Matrix Equations

In the fields of image processing,image restoration,engineering problems,and statistical physics,etc.,there are a large number of problems that rely on solving the positive definite solution of the nonlinear matrix equation X+ATX-1A=Q.With the expansion of research fields,more and more general matrix equations have emerged in many fields,which has aroused great research interest among scholars,such as the matrix equation X-∑i=1m Ai*XAi=Q derived from a class of interpolation problems.This thesis mainly studies the existence conditions of positive definite solutions for generalized matrix equations of the two types of matrix equations mentioned above,X-A*X-nA=I、X+∑i=1m Ai*X-1Ai=Q、X±∑i=1m Ai*X-pi Ai=Q(0<pi≤1)、and matrix equation X-∑i=1m Ai*XAi=Q,as well as the convergence of numerical algorithms for solving these types of matrix equations,and perturbation analysis.The content of this thesis is arranged as follows:In the literature review section,the background,practical significance,and research status of matrix equations are mainly introduced.In chapter 2,we discuss several sufficient conditions for the existence of positive definite solutions for the nonlinear matrix equation X-A*X-nA=I,derives the convergence order of two fixed point iterative algorithms used to solve the equation,and provides perturbation estimates for the positive definite solution.Numerical examples were used to compare the numerical performance of the two fixed point iteration algorithms and inversion-free iteration algorithms,the structure preserving doubling algorithm,and the Newton’s iteration method.The numerical results demonstrate the universality and effectiveness of the fixed point iteration algorithm.In chapter 3,we discuss the sufficient conditions for the existence of positive definite solutions for nonlinear matrix equations X+∑i=1m Ai*X-1Ai=Q.First,we write the equation as an equivalent equation F(X)=X+∑i=1m Ai*X-1Ai-Q=0,and then use the Newton’s iteration method to solve the zero point of this equivalent equation,it is proved that when the initial matrix X0=Q satisfies a certain condition,then the sequence generated by the Newton’s iteration method {Xk}k≥0 converges to the unique positive definite solution of equation X+∑i=1m Ai*X-1 Ai=Q,and the region of the positive definite solution is obtained.In addition,the error estimate of the approximate solution is given.Numerical examples are used to compare the Newton’s iteration method with two commonly used algorithms,and the numerical results demonstrate the superiority of the Newton’s iteration method in terms of convergence speed.In chapter 4,we discuss the existence of positive definite solutions for the nonlinear matrix equation X-∑i=1m Ai*X-pi Ai=Q(0<pi≤1).Using the Thompson metric on the cone,it was proved that there is always a unique positive definite solution to this equation,and an iterative algorithm was designed to calculate the positive definite solution of the equation.In addition,the upper bound of the error estimation for the positive definite solution and the existence intervals of three more accurate positive definite solutions were given.Numerical examples demonstrate the feasibility of the proposed iterative algorithm.In chapter 5,we discuss the nonlinear matrix equation X+∑i=1m Ai*X-pi Ai=I(0<pi≤1),a parameter inversion-free iteration algorithm for solving positive definite solutions of this nonlinear matrix equations is proposed,and its convergence order was proved.Numerical examples have verified the effectiveness of the algorithm and the impact of parameter selection on algorithm performance.In addition,the numerical performance of this algorithm was compared with fixed point iteration algorithm,inversion-free algorithm iteration algorithm,and Newton’s iteration algorithm,the numerical results showed that this algorithm outperformed the inversion-free algorithm and fixed point iteration algorithm in terms of convergence speed,computational time,and computational accuracy;Compared to Newton’s iteration algorithm,it has slightly lower computational accuracy and convergence speed,but less computational time.In chapter 6,we use the differential properties of function matrices to obtain perturbation upper bounds for positive definite solutions of a class of generalized Stein equations X-∑i=1m Ai*X Ai=Q.In chapter 7,we make a summary of our work and propose several issues to be considered in the future.