Convergence Study Of MASOR Iterations

The solution of linear system equation Ax = b is an important part of numerical algebra.It is widely used in practical application such as systems engineering,stability theory,control theory and so on.So the solution of the equation has not only great theoretical significance but also practical value.Usua"lly,there are two methods to solve Ax = b,one is direct method,the ot:her is the iterative method.The direct method is of high computational complexity and not easy to solve,so the method focuses on the iterative method.The convergence is the key to the iterative method,it is an important index to evaluate the performance of specific iterative methods.In recent years,the convergences and the convergence rates of iterative methods have become hot in research.Recently,the main directions of iterative method include AOR iterative method,SOR iterative method,MAOR iterative method,SAOR iterative method,MSOR iterative method and so on.This paper firstly discusses the MASOR iterative method when A is a p-cyclic matrix,then the relationships of the eigenvalues and eigenvectors between the Jacobi and the MASOR iterative methods are studied,finally we give the convergence of MASOR iterative matrix when the coefficient matrix is a 2-cyclic matrix.The structure of this thesis is organized as follows:(1)Introduce the development background and research status.Researchers show that,when dividing the coefficient matrix in different ways,the corresponding iterative methods are also different.In this part,some basic concepts and conclusions of iterative method are introduced firstly,then we divide the coefficient matrix in a proper way and propose the MASOR iterative method,on this basis we get the iterative matrix of MASOR.(2)Establish the relationship between the eigenvalues of the MASOR and the Jacobi iterative method.After that we get the relationship between their eigenvec-tors.(3)Discuss the convergence and convergence rate of the MASOR.We set the coefficient matrix be a 2-cyclic matrix,and get the relationships between the eigenvalues and eigenvectors of MASOR and Jacobi iterative method,accordingly we analyze its convergence and convergence rate when the square of the eigenvalues of Jacobi is positive,negative and pure imaginary number.At last,we get the conclusion that in certain range when Jacobi does not converge,MASOR converges.After that,we give specific examples respectively to verify the theoretical results.