A Kind Of VRP-GMRES(m) Iteration Algorithm For Linear Equations

To solve large scale linear equations discretized through differential equations andintegral equations obtained from the scientific computations and engineering applicationshas become an important subject recently. At present, the Generalized Minimal RESidual(GMRES(m)) iteration algorithm established based on the Galerkin principle was assumedto be the most effective method to solve such linear systems. Therefore, GMRES(m)algorithm and its application in other fields has gradually become the critical topic ofresearchers. Numerical experiment indicate that when the coefficient matrix A is awell-conditioned matrix, GMRES(m) algorithm and some simple improved GMRES(m)algorithm can effectively solve these linear systems. But if the coefficient matrix A wasan ill-conditioned matrix, we should always combine the GMRES(m) algorithm withsome particular pretreatment technology to solve these problems.In actual calculations, once the restart parameter in GMRES(m) and pretreatmentGMRES(m) algorithm is chosen, it will keep fixed in the whole iteration process. So theselection of parameters m is also one of the key factors which influence the effectiveimplementation of the algorithms. For a smaller m may result in slow or even noconvergence and a lager m may cause excessive demand on storage space. In this articlea kind of GMRES(m) algorithm with variable restart parameters was proposed, namely,the VRP-GMRES(m) algorithm. By properly changing the variable restart parameter forthe GMRES(m) algorithm, the iteration stagnation problem resulted from improperselection of the parameter m is resolved efficiently.This paper is organized as follows. In the second chapter, we briefly introduced thefundamental theory of the GMRES(m) algorithm. In the third chapter, an iterative methodnamed VRP-GMRES(m) algorithm is proposed to solve linear equations and Givensorthogonal transformation is used to prove that the proposed algorithm is not only fastconvergent but also is highly accurate. In the fourth chapter, the VRP-GMRES(m)algorithm is applied to solve the linear equations which are obtained from thediscretization of the partial differential equation. It again demonstrated that theVRP-GMRES(m) algorithm is highly effective and accurate. Its superiorities will be much more remarkable when it is used to solve larger scale problems. So it has extensiveprospect in scientific and engineering computing. In the fifth chapter, we firstly study thenormal WGMRES(m) algorithm, then give a new weight factor for this WGMRES(m).And propose a new version of WGMRES(m) by properly changing the parameter m,namely, VRP-WGMRES(m) algorithm. Finally, we give some numerical examples.