Iterative Methods For Several Structured Matrix Problems

In this dissertation,we mainly focus on studying these problems: solving saddle point problems in large scale sparse linear systems,solving coupled algebraic Riccati equations in nonlinear matrix equations,Krylov subspace method using in dimensionality reduction.Details are as follows.Chapter 2,we propose ASOR-like method for solving generalized saddle point problems.First,we propose ASOR-like method for solving generalized saddle point problems based on that for general saddle point problems.By analyzing the properties about eigenpairs of iteration matrix,we give sufficient conditions for the convergence of the ASOR-like method for generalized saddle point problems.By comparing different methods,numerical results show the efficiency of new method.Chapter 3,we propose a modified ASOR-like method for solving saddle point problems.By accelerating the ASOR-like method with two parameters,we introduce a new parameter in ASOR-like method and propose a modified ASOR-like method.Similarly,we analyze the properties about eigenpairs of iteration matrix and give the sufficient and necessary conditions for convergence.Furthermore,we discuss the choice of parameters and verify its rationality in numerical experiments.By comparing with other methods,numerical results show the effectiveness of the modified ASOR-like method.Chapter 4,we discuss Newton's method for coupled discrete-time algebraic Riccati equations.By applying Newton method,solving coupled discrete-time algebraic Riccati equations can be converted into solving coupled Stein equations.We give several iterative methods for solving coupled Stein equations.Apply the structure of iteration matrices,we analyze the convergence of these iterative methods.Based on that,we discuss the solvability and quadratic convergence of Newton's method for coupled discrete-time algebraic Riccati equations.Algorithm is given and numerical experiments illustrate the feasibility of this method.Chapter 5,we discuss Newton's method for coupled continuous-time algebraic Riccati equations.Linearize the coupled continuous-time algebraic Riccati equations by using Newton's method,the problem can be converted into solving coupled Lyapunov equations.For solving coupled Lyapunov equations,we discuss several iterative methods and analyze their convergent properties.Based on the analysis,we further discuss the solvability and quadratic convergence of Newton's method for coupled continuous-time algebraic Riccati equations.Finally,algorithm and illustrative examples are presented to show the effectiveness of the method.Chapter 6,we give an inexact Krylov subspace method for exponential discriminant analysis.We first discuss the efficient computation for matrix exponential-vector products,then show the feasible of inexact strategy by analysing the relationship between the accuracy of approximate eigenvectors and distance to nearest neighbor classifier.We give an inexact Krylov-EDA algorithm based on the theoretical analysis.Moreover,we compare the discriminant criterions of exponential discriminant analysis(EDA)and that of linear discriminant analysis(LDA)from a theoretical point.At last,numerical experiments on some real-world databases verify the theoretical results and show the superiority of new algorithms.