Researches On Four Numerical Methods For Complex Symmetric,Linear Complementarity And Linear Discrete Ill-posed Problems

This dissertation is mainly focused on the numerical solutions for three kinds of large sparse linear systems,namely complex symmetric system of linear equations,linear complementarity problem and linear discrete ill-posed problem.It is of great theoretical value and practical meaning to construct fast and efficient numerical methods for solving these linear systems.In Chapter 2,for solving a class of complex symmetric system of linear equations,we combine the minimum residual technique with the modified Hermitian and skew-Hermitian splitting(MHSS)iteration scheme and propose an iteration method referred to as minimum residual MHSS(MRMHSS)iteration method.Compared with the classical MHSS method,the MRMHSS method involves two more iteration parameters,which can be automatically and easily computed in practical implementations.Then,some properties of the MRMHSS iteration method are carefully studied.Finally,we use four examples to test the feasibility and reliability of MRMHSS iteration method by comparing its numerical results with several other iteration methods.In Chapter 3,for a class of large sparse linear complementarity problem with nonsymmetric positive definite coefficient matrix,by reformulating it as equivalent implicit fixed-point equations,we then establish a highly efficient modulus-based matrix splitting iteration method which will be referred to as MINPS iteration method.This method is made up of an inner iteration and an outer iteration,of which a modulus iteration is used as the outer iteration and the inner iteration applies a preconditioned matrix splitting iteration method with its inexact variant used to solve the module equations at each outer iteration.The convergence properties of the MINPS iteration method are carefully analyzed.By comparing the numerical results of MINPS with those of other existing iteration methods,we illustrate the efficiency and feasibility of our method when used for solving linear complementarity problems.For solving linear discrete ill-posed problems arising in many areas of scientific computation and engineering application,LSQR is one of the most popular numerical schemes,which needs small storage requirement and has well numerical stability.But since the LSQR has semi-convergence,i.e.,too few iterations give an approximate solution that may lack many details that can be of interest,while too many steps yield an approximate solution that suffers from a large propagated error due to the error in the data,thus it is important to terminate the iterations after a suitable number of steps.In Chapter 4,we further study the LSQR iteration method with the aid of proposing a simple but efficient stopping criterion,precisely speaking,which is based on comparing the residual errors associated with iterates generated by the LSQR and Craig iterative methods to determine the regularization parameter.A great many numerical results show that the LSQR can perform well when applied to solve the practical problems of which the variance of the errors are not known.In Chapter 5,we again consider the above-mentioned linear discrete ill-posed problems,whose computing solutions are significantly sensitive to perturbations in the data and we usually apply a regularization method to reduce the sensitivity in practice.Based on an iterated Tikhonov regularization method(AIT)proposed by Donatelli and Hanke(2013),in which the original coefficient matrix is approximated by a proper matrix that is easier to work with,in this method the amount of computations are decreased and this method works well in some practical examples.Nevertheless,its convergence condition is rarely satisfied in practice and the computed solution is sensitive to the perturbation of the measured data.Thus,we propose a more stable iteration method called MAIT for solving linear discrete ill-pose problems,moreover,the theoretical properties and convergence results are studied in detail.From numerical experiments we even find that MAIT works more widely than AIT,specifically,when the noise level of measured data is low,AIT method fails,however,MAIT method can still efficiently solve this kind of problems.