Study On Some Problems Of Matrix

In this dissertation,we mainly study the following problems: solving complex system of linear equations,solving generalized saddle point problems,solving inverse singular value problems and parameter estimation problems of multiple restricted partitioned linear model,details are given as follows:.Chapter 2,we introduce a new parameter to generalize the single-parameter CRI iteration method and obtain a generalized CRI iteration method to solve complex symmetric system of linear equations.The convergence conditions of this method are discussed.Meanwhile,an upper bound of the spectral radius of the iteration matrix is introduced,and the values of the parameter when the upper bound is minimized are given.Finally,some numerical experiments are given to illustrate the effectiveness of the generalized CRI method.Chapter 3,a relaxed block splitting preconditioner is proposed to solve complex symmetric indefinite system of linear equations by using relaxation techniques.We study the eigenvalue distributions and the properties of the corresponding eigenvectors of the preconditioned matrix and some numerical experiments are given to prove the effectiveness of the preconditioner.Chapter 4,for nonsingular generalized saddle point problems,we construct a class of two-parameter matrix splitting preconditioner and analyze the eigenvalue distributions of the preconditioned matrix with the change of the parameters.It is proved that the eigenvalue distributions of the preconditioned matrix become more and more concentrated as the two parameters tend to be more smaller,and they are clustered around two points.Finally,some numerical experiments are given to verify our theoretical analysis results.Chapter 5,based on QR decomposition and Newton's method,a algorithm is proposed to solve inverse singular value problems.According to the structure of the matrix,the algorithm is improved by using rank-revealing technique.Then the convergence of the algorithms is analyzed.Finally,some numerical experiments are given to describe the convergence results of the algorithms.Chapter 6,parameter estimation problems under a partitioned linear model with two constraint conditions are extended to the study of parameter estimation problems under a multiple partitioned linear model with s constraint conditions.We discuss the relationships of the best linear unbiased estimators between the multiple restricted partitioned linear model and the corresponding s small restricted linear models.Meanwhile,some statistical properties of these parameter estimations are also discussed.