A note on relaxed and flexible GMRES

We consider the solution of a linear system of equations using the GMRES iterative method. In [3], a strategy to relax the accuracy of the matrix-vector product is proposed for general systems and illustrated on a large set of numerical experiments. This work is based on some heuristic considerations and proposes a strategy that often enables a convergence of the GMRES iterates xk within a relative normwise backward error less than a target accuracy. Finally a significant step toward a theoretical explanation of the observed behaviour of the relaxed GMRES is proposed in [16, 17]. In these works, important justifications are brought to the fact that a relaxation of the matrix-vector product proportional to the inverse of the norm of the residual may enable the convergence of the relaxed GMRES. In this paper we extend these works, we establish a computable relaxation strategy that enables to reach the aims of [3] using the tools presented in [16, 17]. We investigate the compliance of our strategy with the scaling invariance properties of GMRES. We extend the study to the inexact preconditioning situation and explore relationships with Flexible GMRES. We report results on intensive numerical experiments to illustrate the behaviours of the relaxed GMRES monitored by the proposed relaxation strategy. Finally in the case of the Householder relaxed GMRES we establish a backward stability result by extending the results of [5].