Backward Iterative Methods Based On The New Pre-conditions For Sub-under Analysis

When we numerically simulate the practical problems in the natural sciences and engineering computations, one ultimately brings in solving the large sparse matrix of linear algebraic equations, such as the development of oil and gas resource, the simulation of nuclear explosions , the weather reports, and numerical wind tunnel etc... Solving the large linear equations obtained from the discrimination for the elliptic partial differential equation is always a concerning subject, and the Jacobi iterative method, the Gauss-Seidel iterative method, the SOR and JOR method have been presented to provide us great convenience. However sometimes, the convergence of these classic methods are slow or even not convergence, this motivates people to seek the high speed convergence of the iterative method.In recent years, solving the linear equations technique has been greatly developed, especially in the emergence of the pre-condition technology. Many researchers have investigated the various classic pre-conditioned iterative methods when the coefficient matrix is special (such as the diagonally dominant Z-matrix, or Q-matrix). In the pre-condition technology, it is crucial to find the suitable pre-conditioner. In reference [1]-[6], many scholars obtained some important theories under the various coefficient matrixes and the different pre-conditioners.This paper is to propose a new pre-conditioner using the backward iterative method based on the theories of the former works, with the new pre-conditioner the method not only proves its convergence when the coefficient matrix A is non-singular M-matrix and H-Matrix, but also acquires the BIMSOR, BIMGS's convergence speeds are significantly faster than those of the classic BSOR, BGS iterative methods. By choosing the special values of the pre-condition factors, the convergence speed of the block iterative methods are much faster than the classic BSOR and BGS iterative methods, and we apply the new method to the practical problems and demonstrate the superiority and robustness of the pre-conditioned iterative method.The following is the structure and main content of this paper.The first part is the introduction. We introduce the background for the linear algebraic equations and the pre-conditioned method, and outline the iterative matrix followed from the Jacobi, Gauss-Seidel, BSOR, and BAOR iterative method.The second part is the preliminaries. It mainly gives some essential notations, definitions and lemmas for the M-matrix, comparison matrix, splitting matrix and so on.The third part is the prior knowledge. In this part, we give the conclusion that have been proposed and review the predecessors'works about the pre-conditioner, thus present the idea how to construct the new pre-conditioner of this paper.The fourth part is the main conclusions. Under the assumption that the coefficient matrix A has some peculiarities we present the convergence of the BIMSOR, BIMGS methods, and demonstrate these methods are much faster than the classic Gauss-Seidel, BSOR iterative methods to prove the theoretical convergence rate, and the numerical results in every part are verified the main conclusions convincingly.The fifth part is the summary and prospect. This part mainly summarizes the main idea and conclusion in this paper, and takes the prospect on the pre-conditioned methods for the future.