This paper studies the fundamental theory of the generalized minimal residual algorithm(GMRES(m))in Krylov subspace and specially the relationship between residual vector and Krylov subspace.The relationship of the algorithm convergence and the subspace be selected is further researched according the linear system about residual vector.It is posed that the convergence can be slowed down because there are so many very small eigenvalue in magnitude.And a accelerated method(AGMRES(m)) is proposed to improve the convergence of the GMRES(m).Theoretical analysis and numerical results show the reliability and efficiency of the algorithm.
