Several Iterative Algorithms For Solving Sylvester Matrix Equation

Sylvester matrix equation has been widely applied in many fields of scientific computing and engineering technology,such as dynamic system,control theory,system theory,image restoration,signal processing,the optimal approximation problem of matrix,neural network,model reduction,and so on.Because of the wide application background of Sylvester equation,it is of great theoretical and practical significance to solve Sylvester matrix equation quickly and effectively.In this thesis,we construct several iterative algorithms for solving Sylvester matrix equation (3 + (3 = ,based on some existing achievements.Moreover,we also analyze the convergence of the algorithms and further verifies the effectiveness of the proposed algorithms through numerical experiments.The main contents of this thesis are organized as follows:In Chapter 1,the application background and research status of Sylvester matrix equation and some related preliminary knowledge are introduced,and the main contents of this thesis are also introduced.In Chapter 2,we propose a parameterized single step HSS iterative algorithm(PSHSS)for solving the Sylvester matrix equation by extending the parameterized single step HSS iterative algorithm for solving linear equations to matrix equations.Firstly,we introduce the construction process of PSHSS iterative algorithm.Then,the convergence of the algorithm is analyzed.Finally,numerical experiments verify the feasibility and effectiveness of this algorithm.In Chapter 3,we propose an Euler extrapolation iterative algorithm(EE)for solving Sylvester matrix equation,by combining Hermitian and skew-Hermitian split iterative methods with Euler formula.Firstly,we introduce the basic idea and iterative form of the algorithm.Then,the convergence of the algorithm is analyzed theoretically in detail.Finally,some numerical experiments further show that the effectiveness of the proposed algorithm.In Chapter 4,we establish a single step scale iterative algorithm(SDSS)to solve the Sylvester matrix equation by combining the double step scale splitting iterative method with the single step HSS iterative method.Firstly,we give the basic idea of this method and the corresponding iterative algorithm.Then,the convergence of the algorithm is analyzed.Finally,numerical experiments further verify the effectiveness and feasibility of the proposed algorithm.In Chapter 5,we sum up the research content of this paper and put forward the future research direction.