Implementation of the GMRES Method Using Householder Transformations

Abstract

The standard implementation of the GMRES method for solving large nonsymmetric linear systems involves a Gram-Schmidt process which is a potential source of significant numerical error. An alternative implementation is outlined here in which orthogonalization by Householder transformations replaces the Gram-Schmidt process. This implementation requires slightly less storage but somewhat more arithmetic than the standard one; however, numerical experiments suggest that it is more stable, especially as the limits of residual reduction are reached. The extra arithmetic required may be less significant when products of the coefficient matrix with vectors are expensive or on vector and, in particular, parallel machines.