L-matrix Preconditioned Iterative Method Convergence Analysis

The L-matrix is an actual context of a broad category matrix, as we all know, the large scale sparse linear systems often appear in a wide variety of areas of mathematics, physical, fluid dynamics and economics science. So, solving efficiently these systems aroused many authors'interests. The iterative method which can take full advantage of the sparse matrix, thereby saving memory cell, so it is a more practical way to solve large sparse linear algebraic equations. The rule whether the iterative is good is usually described by convergence and convergence rate, Thus, we should find an iterative method which has good convergence and fast convergence rate, this owns practical value. In order to solve linear system faster and better, we quote nonsingular preconditioner. By preconditioner, we accelerate the convergence rate of iterative method. Based on [1] to [3], this paper proposes two preconditioner, and assumed the coefficient matrix is an irreducible nonsingular L-matrix. By comparing Spectral radius, we discuss the convergence rate between the preconditioned iterative methods and the basic iterative method, whether this results are true or not for general preconditioner? In this paper, we also consider the general preconditioner, and get the comparison theorem between the preconditioned iterative methods and the basic iterative method, which generalize and improve the original conclusion.The followings are the construction and main contents of this paper:The first part is introduction. We give the iterative matrix of basic SOR and Gauss-Seidel iterative method, as well as the background of preconditioned iterative method, and then we introduce the preconditioner P and the iterative matrix of preconditioned iterative method.The second part is preliminaries. In this part, some important definition and lemma which is required in this paper are given.The third part is the conclusions have been proposed. In this part, we mainly introduce the preconditioned iterative method proposed by predecessors, including how to select the preconditioned matrix and the comparison theorems relatively.The fourth part is comparison theorem, is the main part of this paper. In this part, we assume the coefficient matrix is an irreducible nonsingular L-matrix, then discuss the convergence rate between preconditioned iterative method and basic iterative method, and also consider the general preconditioner. We get the convergence rate of preconditioned iterative method is faster than the basic iterative method.The fifth part is numerical examples. In this part, we give the numerical examples, which show the effectiveness of our main results in part four.The sixth part is summary and prospect. In this part, we summarize the main ideas,method and the main results which are given in this paper, then we prospect the development of the preconditioned iterative methods in the future.