Using successive approximations for improving the convergence of GMRES method

summary:In this paper, our attention is concentrated on the GMRES method for the solution of the system $(I-T)x=b$ of linear algebraic equations with a nonsymmetric matrix. We perform $m$ pre-iterations $y_{l+1}=Ty_l+b $ before starting GMRES and put $y_m $ for the initial approximation in GMRES. We derive an upper estimate for the norm of the error vector in dependence on the $m$th powers of eigenvalues of the matrix $T$. Further we study under what eigenvalues lay-out this upper estimate is the best one. The estimate shows and numerical experiments verify that it is advisable to perform pre-iterations before starting GMRES as they require fewer arithmetic operations than GMRES. Towards the end of the paper we present a numerical experiment for a system obtained by the finite difference approximation of convection-diffusion equations