Preconditioned Iterative Method And Error Estimates Of Linear Equations

The thesis mainly discusses several different preconditioned iterative methods and a new form of error estimate for the AOR iterative method. As we all know, many mathematical problems, which have turned up in the fields of computer science and engineering physics can be attributed to how to get the solutions of the linear system of equations. In order to solve the linear system of equations, two methods are usually adopted, that is, the direct method and the iterative method. With the development of information technology, many advantages have been found for the iterative method, whose corresponding programming is simple and practicable. Meanwhile, the coefficient matrix of the linear system of equations can be unchanged under the computation by the iterative method. Since convergence is the core issue for the iterative method, it is meaningful to study how to accelerate the rate of convergence under this circumstance. Having used different preconditioners for the iterative method, convergence can be accelerated, even some unconvergent iterative methods can be improved to be convergent by many scholars. In the present thesis, the problem of convergence for the preconditioned iterative method is studied under the condition that the coefficient matrix A of the linear system of equations are L matrix or H matrix.The linear system of equations all have the form of Ax= b, where the coefficient matrix A is an n×n real matrix, b is a given column vector in Rn, the solution vector x exists in Rn. The thesis consists of three parts, the Preface, Chapter One and Chapter Two.The Preface mainly discusses about several types of classical preconditioning matrices and other preliminaries.The Chapter One, which plays an important part in this thesis. Based on the preconditioning matrices Ps1 and Ps2 proposed by Evans et al, the convergence of preconditioned Gauss-Seidel iterative method is best is derived as main result. In addition, based on the preconditioning matrix P1(α) proposed by T.kohno et al in 1997 and P2(α) introduced by A.Hadjidimos et at in 2003, the thesis constructs a new preconditioning matrix P2 by changing the value range of parameters and the place of nonzero elements not in the diagonal line of the matrix. Moreover, considering P2 as the preconditioning matrix and the H matrix as the coefficient matrix, it studies preconditioned AOR, SOR and Gauss-Seidel iterative methods, and points out that the preconditioning matrix P2 is valid by setting numerical examples for support.In Chapter Two, a new form of error estimate for the AOR iterative method is presented, based on the one corresponding to M-1N, where M is a doubly diagonal dominating matrix and N is an n×m matrix. Especially, whenγ=ω= 1, a simple error estimation for Gauss-Seidel iterative method is obtained and numerical example is given for support.