Convergence behaviour of deflated GMRES(m) algorithms on AP3000

Abstract

GMRES(m) method, the restarted version of the GMRES (generalized minimal residual) method, is one of the major iterative methods for numerically solving large and sparse nonsymmetric problems of the form Ax=b . However, the information of some eigenvectors that compose the approximation disappears and then the good approximate solution cannot be obtained, because of this restart. Recently, in order to improve such a weak point, some algorithms which named MORGAN, DEFLATION and DEFLATED-GMRES algorithm, have been proposed. Those algorithms add the information of eigenvectors that can be obtained in the previous restart frequency. In this paper, we study those algorithms and compare their performances. From the numerical experiments on the distributed memory machine Fujitsu AP3000, we show that DEFLATED-GMRES( m, k ) method performs the good reduction of residual norms in these algorithms.