Scalable parallel implementations of the GMRES algorithm via Householder reflections

Applications involving large sparse nonsymmetric linear systems encourage parallel implementations of robust iterative solution methods, such as GMRES(k). One variation of GMRES(k) is to adapt the restart value k for any given problem and use Householder reflections in the orthogonalization phase to achieve high accuracy. The Householder transformations can be performed without global communications and modified to use an arbitrary row distribution of the coefficient matrix. The effect of this modification on the GMRES(k) performance is discussed here. This paper compares the abilities of various parallel GMRES(k) implementations to maintain fixed efficiency with increase in problem size and number of processors.