A Convergent Restarted GMRES Method For Large Linear Systems Minghua Xu2,Jinzi Zhaol,Jiancheng Wu2 and Hongjun Fan 1.State Key Laboratory for Novel Software Technology,Nanjing University,Nanjing 210093,P.R.China 2.Department of Information Science,Jiangsu Polytechnic University,Changzhou,213016,P.R.China October 3,2003 Abstract The GMRES method is popular for solving nonsymmetric linear equations.It is generally used with restarting to reduce storage and orthogonal- ization costs.However,it is possible to show that the restarted GMRES method may not converge,i.e.,it may be stationary.To remedy this difficulty,a new convergent restarted GMRES method is discussed in this paper. Key words GMRES,Krylov subspace,iterative methods,nonsymmetric sys- tems. AMS(MOS)subject classifications 65F10 1.Introduction The restarted GMRES algorithm GMRES(m)1]proposed by Saad and Schultz is one of the most popular iterative methods for solving large linear systems of equations Ax=b,A∈Rnxm,x,b∈R, (1.1) with a sparse,nonsymmetric,and nonsingular matrix A.It is known that when A is positive real,the restarted GMRES method will produce a sequence of ap- proximates tk that converge to the exact solution.However,when A is not positive real,this method often slows down convergence and stagnates.The analysis and implementation of the restarted GMRES algorithm continue to re- ceive considerable attention245.6.7..For example,Y.Saad suggested a flexible inner-outer preconditioned GMRES method FGMRES(m).R.B.Morgan gave a restarted GMRES method augmented with eigenvectors ],and Cao Zhihao et. al.presented a convergent restarted GMRES algorithm based on the algorithm FGMRES(m)14l.We will now briefly review the algorithm GMRES in this sec- tion.A new restarted GMRES method and its analysis will be given in section 2,section 3 gives the examples and comparisons,and conclusions are given in section 4.The restarted GMRES can be briefly described as follows. Algorithm 1:GMRES(m)for systems (1.1)A Convergent Restarted GMRES Method For Large Linear Systems M inghua Xu1,2 ,J inxi Zhao1 ,J iancheng W u2 and Hongjun F an1 1.State Key Laboratory for Novel Software Technology ,Nanjing University, Nanjing 210093,P.R.China 2.Department of Information Science, Jiangsu Polytechnic University,Changzhou,213016,P.R.China October 3, 2003 Abstract The GMRES method is popular for solving nonsymmetric linear equations. It is generally used with restarting to reduce storage and orthogonal￾ization costs. However, it is possible to show that the restarted GMRES method may not converge, i.e., it may be stationary. To remedy this difficulty, a new convergent restarted GMRES method is discussed in this paper. Key words GMRES, Krylov subspace, iterative methods, nonsymmetric sys￾tems. AMS(MOS) subject classifications 65F10 1. Introduction The restarted GMRES algorithm GMRES(m)[1] proposed by Saad and Schultz is one of the most popular iterative methods for solving large linear systems of equations Ax = b, A ∈ R n×n , x, b ∈ R n , (1.1) with a sparse, nonsymmetric, and nonsingular matrix A. It is known that when A is positive real, the restarted GMRES method will produce a sequence of ap￾proximates xk that converge to the exact solution. However, when A is not positive real, this method often slows down convergence and stagnates. The analysis and implementation of the restarted GMRES algorithm continue to re￾ceive considerable attention [2,3,4,5,6,7,8]. For example, Y.Saad suggested a flexible inner-outer preconditioned GMRES method FGMRES(m)[2]. R.B.Morgan gave a restarted GMRES method augmented with eigenvectors [3] ,and Cao Zhihao et. al. presented a convergent restarted GMRES algorithm based on the algorithm FGMRES(m)[4]. We will now briefly review the algorithm GMRES in this sec￾tion. A new restarted GMRES method and its analysis will be given in section 2, section 3 gives the examples and comparisons, and conclusions are given in section 4. The restarted GMRES can be briefly described as follows. Algorithm 1: GMRES(m) for systems (1.1) 1