The Convergence Analysis Of Matrix In The New Pre-conditions And Its Comparison

Many problems in mathematics, physics, hydrodynamics, engineering and economics are attributed to solving the large-scale sparse linear algebraic equations. The iterative method for the algebraic equations can take full advantage of the sparsity of the matrix to save the computer memory space, so that the iterative method plays an important role in computational problems, and becomes an more practical approach for solving the large-scale sparse linear algebraic equations. It is no practical value for the non-convergence or the slow speed convergence of the iterative method, and what works is the iterative method with good and faster convergence, which has the realistic meanings.In order to solve these large sparse linear equations more accurate and faster, people proposed several methods including Jacobi iterative method, Gauss-Seidel iterative method and the basic iterative methods including SOR, JOR and so on with the introduction of the relaxation factor and the acceleration factor. In recent years there is a great development in solving the algebraic equations, especially after the introduction of the pre-condition matrix, and the iterative method's convergence rate accelerates greatly, which satisfies people's computational demands.In this paper we propose a new pre-condition matrix of I + Son the former works of the predecessors, and when the coefficient matrix is non-singular irreducible M-matrix and H-matrix respectively we discuss the convergence analysis of not only the pre-conditioned SOR (denoted as PSOR) method rather than the classical SOR method, but also the pre-conditioned JOR (denoted as PJOR) method rather than the classical JOR method. We prove their convergence results under the new pre-conditions, and demonstrate the convergence rate of PSOR and PJOR is clearly much faster than that of the classical SOR and JOR method, which show the superiority of our new pre-condition iterative method in this paper.The construction and main content of this paper is as follows.The first part is introduction. We give the background of pre-conditioned methods and introduce the classical SOR, JOR iterative methods, and give the pre-condition iterative matrix P of the iterative method of PSOR and PJOR, respectively.The second part is preliminaries and related results, we introduce some important definitions and lemmas about M -matrix, H -matrix, and matrix splitting and so on,and some important results that the predecessors have done on the pre-condition methods in recent years, and thus put forward the new pre-conditioner in this paper.The third part is one part of the main conclusions, under the assumption of the coefficient matrix is non-singular irreducible M-matrix and H-matrix we illustrate the convergence of the iterative methods under the action of a new pre-condition matrix.The fourth part is the other part of our main conclusions, in this part we convince the convergence of the pre-conditioned SOR method and the pre-conditioned JOR method on the basic of the third part, and compare these with the classical SOR and JOR methods, and we conclude that with the pre-condition our methods'convergence rate is faster than that of the classical ones. Furthermore, we verify the correctness of our conclusions through the numerical examples.The fifth part is the final conclusion, we summarize the major ideas and conclusions in the paper, and take a prospect for the development of the pre-conditioned iterative method.