Research Of New Preconditioned Iterative Methods On Linear Equations

Many of the problems we encounter in real life at the time of calculation, and finally into a linear equation of the form, such as Ax=b. In order to better and faster to solve linear equations Ax=b, in which the iterative method is a more effective method. Convergence speed of iteration method can be used on the size of the spectral radius of iterative matrix to describe. We know that the necessary and sufficient conditions for convergence of first-order stationary iterative method:the spectral radius of iteration matrix is less than1, so that we should look for an iterative matrix spectral radius of a relatively small iterative method. In fact, for this purpose, we usually by the pretreatment method to accelerate the convergence rate of iterative methods.This article first describes the classic iterative method for solving linear equations. On this basis, introducing the pretreatment matrix P=1+Wβ, proposed for solving linear equations of the new preconditioned Gauss-Seidel iterative method and preconditioned AOR iterative method. On the assumption that the linear equation coefficient matrix is singular system of linear equations diagonally Z-matrix,H-matrices and non-singular irreducible M-matrix case, apply the new preconditioned iterative methods, get the convergence theorems of iterative method and comparison theorem. Finally a numerical example is given to illustrate:select the appropriate preconditioned factor can make the solution of linear equations of the new preconditioned iterative method becomes more superior.