A Weighted Simpler GMRES Algorithm

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,such as Simpler GMRES and Weighted GMRES. Simpler GMRES converts a least squares problem into a triangular matrix linear system,so it is less expensive than GMRES.Weighted GMRES adds a weighted technique to speed up the convergence,but become more expensive.In this paper, we give a new method called Weighted Simpler GMRES,which is based on Simpler GMRES and is obtained by combining it with Weighted GMRES.The Numerical experiments show that for many problems,Weighted Simpler GMRES converges faster than Simpler GMRES,Weighted GMRES and GMRES,and is less expensive than Weighted GMRES.This paper includes four parts.In the first part,related problems and baekground is introduced and the main contents of this paper are also described.In the second part,we briefly describe the GMRES,Simpler GMRES and Weighted GMRES.In the third part,the main idea of Weighted Simpler GMRES is introduced and some related properties are discussed too.The last part is the numerical experiment.We test a wide range of problems to show that Weighted Simpler GMRES has better performance than some original algorithms.