Some Researches On Numerical Algorithms For Solving Several Class Of Sylvester Matrix Equations

With the development of science and technology,different matrix equations are derived driven by a variety of practical problems,such as Lyapunov equations and Sylvester equations,the study of them will push related disciplines to the deeper development.Sylvester equations play an important role in control theory and many other fields of engineer.Because of their vital applications,Sylvester equations have attracted considerable attention from many researchers.In recent decades,the theory of matrix equations has been extensively studied both at home and abroad.Therefore,looking for quick and effective method of the Sylvester equations is particularly necessary.In this thesis,the theoretical analysis and numerical algorithm have been studied in detail for several kinds of matrix equations in control theory,then we obtained the corresponding numerical results.The main content of this thesis is as follows:In the introduction,the background and applications of several kinds of equations are introduced,such as Lyapunov equation and Sylvester equation.The development and research status of Sylvester equation are discussed briefly.In Chapter 1,the parameter iteration method for obtaining the unique solution of a kind of Sylvester matrix equation is studied.First of all,we introduce the parameter iteration method to solve the linear equation and the iteration-correction method of Sylvester matrix equation.Then,the convergence analysis has been done for the parameter iteration method,and we give the optimal parameters and approximate optimal parameters.Through the analysis of partial sum,we establish the accelerated algorithm for parameter iteration method.Some numerical experiments are given to show that the method is effective.In Chapter 2,we establish a modified conjugate gradient(MCG)algorithm for solving the Hermitian R-conjugate solutions of the generalized coupled Sylvesterconjugate equation.Firstly,we introduce the MCG algorithm for solving the generalized Sylvester-conjugate matrix equation.Then,the algorithm is convergent within finite iterative steps in the absence of round-off error for any initial given matrix.Furthermore,some numerical examples demonstrate the effectiveness of this method.In Chapter 3,on the basis of the HSS iterative method,a generalized parameterized Hermitian and skew-Hermitian splitting iteration method for solving Sylvester matrix equation is studied.Then by analyzing the correlation properties of spectral radius,the quasi-optimal values of the iteration parameters for the GPHSS method is also obtained.Finally,the feasibility of this method is verified by numerical experiments.