Imgs Method For A Class Of Convergence And Comparison Theorems,

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 more better, we quote nonsingular preconditioned matrices. by preconditioned matrices, we accelerate the convergence rate of iterative method. Based on [1] to [6], this paper proposes the IMGS iterative method, which has preconditioned matrix P = I + Sα.firstly, we discusse the convergence of the IMGS iterative method for the coefficient matrix is the nonsingular M -matrix,H -matrix and strict diagonal dominant matrix. Then assumed the coefficient matrix is an irreducible nonsingular M -matrix, we discuss the convergence rate between the IMGS iterative methods and basic TOR iterative method,PSOR iterative method,PAOR 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 background of preconditioned iterative method, as well as the iterative matrix of basic TOR,SOR and AOR iterative method, then we introduce the preconditioned matrix P and the iterative matrix of IMGS,PSOR and PAOR iterative method Separately.The second part is preliminaries. This part mainly make preparations for part four and five. In this part, some important definition and lemma are given, for example, M - matrix,H -matrix,matrix splitting and so on.The third part is the conclusions have been proposed. In this part, we mainly intro- duce the preconditioned iterative method proposed by predecessors, including how to s- elect the preconditioned matrix and the comparison theorems relatively.The fourth part is the convergence of IMGS iterative method, is also the main part of this paper. In this part, we discuss the convergence of IMGS iterative method, and we proved the IMGS iterative method convergence if the coefficient matrix is M -matrix,H -matrix and strict diagonal dominant matrix .Then the numerical example given illus- trate the main result.The fifth part is comparison theorem, is also the main part of this paper. In this part, firstly we assume the coefficient matrix is an irreducible nonsingular M -matrix, then discuss the convergence rate between IMGS iterative method and basic TOR iterative method,PSOR iterative method,PAOR iterative method. We get the convergence rate of IMGS iterative method is not only faster than the basic TOR iterative method, but also faster than the PSOR iterative method and PAOR iterative method. Finally, a numerical example illustrate the main results.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 develo- pment of the preconditioned iterative methods in the future.