FOM/GMRES and variants

Abstract

FOM (Full Orthogonalization Method) and GMRES (Generalized Minimal RESidual) is a pair of Q-OR/Q-MR methods using an orthonormal basis for the Krylov subspaces. In fact, as we have seen in Chapter 3, the FOM residual vectors are proportional to the basis vectors. Thus, FOM is a Q-OR method for which the residual vectors are orthogonal to each other. It is a true OR (Orthogonal Residual) method. In GMRES, since the basis is orthonormal, the norm of the quasi-residual  is equal to the norm of the residual. Therefore GMRES is a true MR (Minimum Residual) method. The minimization of the norm of the residual implies that in GMRES the residual norms are decreasing which is a most wanted property for iterative methods. However, as we will see, the GMRES norms may stagnate.