Estimates of the Norm of the Error in Solving Linear Systems with FOM and GMRES

We provide formulas for the norm of the error when solving non symmetric linear systems with the full orthogonalization method (FOM) and the generalized minimum residual method (GMRES) as well as relations between the error norm and the residual norm. From these formulas we are able to compute estimates of the norm of the error during the iterations. Since stopping criteria based on the norm of the residual may sometimes be misleading, such estimates could lead to a more robust way to stop the iterations. Numerical experiments show that the proposed norm estimates work nicely on difficult linear systems.