Restarted Simpler GMRES Augmented With Approximate Errors

GMRES is the most popular method for solving large scale nonsymmetric sparse linear systems. There exist a large variety of modifications to the standard GMRES algorithm and accelerating technique is one of them. Augmented methods are an important class of accelerating techniques. They append certain vectors to the current approximation space to speed the convergence of restarted GMRES. LGMRES is a new kind of augmented method. It appends approximate erros to the current approximate space, effectively preventing the alternating behavior for every other residual , which results in slow convergence for the case of GMRES. LGMRES is easy to implement, requiring small changes to the standard GMRES algorithm and it applies to a wide range of problems. In this paper we present a simpler form of LGMRES by simpler Arnoldi process. This is based on the fact that the residual vectors at the end of each restart cycle of restarted simpler GMRES also alternate direction in a cyclic fashion, similar to the case for restarted GMRES. The new variant presented in this paper can prevent the alternating phenomenon and accelerate the convergence rate for the case of simpler GMRES. It requires less amout of work than restarted LGMRES and the numerical experiments show that it has better performance.This paper includes four parts. In the first part, related problems and background is introduced and the main contens are also described. In the second part, we briefly describe the GMRES and simpler GMRES method. In the third part, the main idea of LGMRES is introduced and the new variant, reffered to as simpler LGMRES, is presented. The last part is the numerical...