Convergence Analysis Of Pre-conditions For Gauss-seidel Iteration Method

The solution of large-scale linear equations is a sore of the large-scale scientific and engineering computing. With the rapid development of computers, the direct method has been replaced by iterative method for solving large-scale linear equations and it has been become one of the most important methods. The standard which judges iterative process is usually described by convergence convergence and convergence rate ,thus,we should find an iterative method which has good convergence and fast convergence rate,this owns practical value. Therefore, we should look for iterative methods with the faster convergence speed, so they will have practical value. In many cases, the speed of iteration is described by the spectral radius of the iteration matrix. In this article we describe convergence rate by comparing the spectral radius of iterative matrix.In order to solve linear equations better and faster, here we introduce a non-singular matrix, by the pre-conditioned matrix, we accelerate the convergence rate of iterative method. The comparison theorem in this article is more general than before, it reduces the prerequisite for the establishment of comparison theorem, with the coefficient matrix A of linear equations changing from irreducible diagonally dominant Z -matrix to non-singular M -matrix,so the application of preconditioned comparison theorem is expanded. In this paper, we give three preconditioned matrix, respectively P1 = I + Kβ, PB = I + Band P2 = I + Sm, when the coefficient matrix is non-singular irreducible M-matrix, we discuss the comparison theorems between the pre-conditions for Gauss-Seidel iterative method in the premise of the three pre-conditions of and the classical Gauss-Seidel iterative method, which generalize and improve the conclusions. The first part is the introduction. We give the iterative matrix of classical AOR iterative method, SOR iterative method and the classical Gauss-Seidel iterative method, and introduce the preconditioned matrix P .The second part is preliminaries, and it mainly gives some important definitions and lemmas, such as M -matrix, matrix splitting.The third part is comparison theorem in the preconditioned matrix P1 = I + Kβand it is one of the main conclusions of this article. This part is divided into three parts, firstly we introduce preconditioned matrix, and then give some relevant conclusions, mainly the work done by predecessors. Further more, when the coefficient matrix is non-singular irreducible M -matrix, we discuss convergence rate between preconditioned Gauss-Seidel iterative method with the classical AOR iterative method,preconditioned AOR iterative method,preconditioned SOR iterative method.In the fourth part, we give comparison theorem of preconditioned Gauss-Seidel iterative method in the premise of preconditioned matrix PB = I + B. Similarly, we give preconditioned matrix ,and by comparison we get the convergence rate of preconditioned Gauss-Seidel iterative method is faster than the classical Gauss-Seidel iterative method.Part V gives comparison theorem of preconditioned Gauss-Seidel iterative method in the premise of preconditioned matrix P2 = I + Sm. We mainly talk about the comparison theorem between the preconditioned Gauss-Seidel iterative method with the preconditioned SOR iterative method.The sixth part is a numerical example, which illustrates the main results in this paper.