Two Kinds Of New Preconditioned Iterative Method And Comparison Of Convergence

As we all know, for solving these large sparse linear algebraic equations, Cramer principle can solve its unique solution, but which is not advisable in the practical calculation. Iterative method is favored by many scholars, the necessary and sufficient conditions of iterative method is spectral radius of iterative matrix less than1. When spectral radius of iterative matrix of large linear equations is closing to1, the convergence of classic iterative method is slowly. At last, we give preconditioned matrix is effective way of solve this problem, the convergence rate of preconditioned method is more faster than the classical iterative method. This paper propose the two new preconditioned iterative method based on the theories of the former works, the new preconditioned the method not only proves its convergence when the coefficient matrix A is M-matrix and H-matrix, but also proves that the convergence speeds are significantly faster than those of the classic iterative method and existing iterative methods.First of all, giving the background of preconditioned iterative method, as well as the basic iterative method (Gauss-Seidel iterative method、Jacobi iterative method、SOR iterative method、AOR iterative method), and introducing the development of preconditioned iterative method and some conclusions.Then the next chapter is preconditionedI+B iterative method. We discuss the preconditioned Gauss-Seidel iterative method and preconditioned AOR iterative method is convergent and faster than those of the classic iterative method when the coefficient matrix is M-matrix; then discussing the comparison theorem between the new preconditioned I+B iterative method and the existing preconditioned I+C’ and I+S’ iterative method. At last, we give the numerical examples, which show the effectiveness of our main results.Finally, that chapter is preconditioned I+K iterative method. Discussing the preconditioned Gauss-Seidel iterative method and preconditioned AOR iterative method is convergent when the coefficient matrix is H-matrix; giving the preconditioned I+K iterative method is more faster than the rate of the comparison of preconditioned I+K matrix and comparison matrix; At last, a numerical example is given, which show the effectiveness of main results and faster than those of the classic iterative methods and existing iterative methods.