Using successive approximations for improving the convergence of GMRES method
Jan Zítko · Applications of Mathematics · 1998
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 =T yl +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.