Influence of Residual Vector and Krylov Subspace on Convergence Velocity of GMRES Algorithm

By analysing the structure of restarted GMRES algorithm,we discover that residual vector and Krylov subspace have an influence on the convergence velocity of GMRES(m) algorithm,and deduce that the residual vector has a direction cosine relationship with the first vector and m+1 vector in Krylov subspace.A numerical example is used to verify its rationality.Algorithm analysis indicates that the convergence velocity of GMRES(m)method is slower when the project of the residual vector rk+1 is large in the krylov subspace v1 vector and small in the krylov subspace vm+1 vector,or vice versa.