RESIDUAL SMOOTHING TECHNIQUES FOR ITERATIVE METHODS

An iterative method for solving a linear system Ax = b produces iterat es {x(k)} with associated residual norms that, in general, need not de crease ''smoothly'' to zero. ''Residual smoothing'' techniques are con sidered that generate a second sequence {y(k)} via a simple relation y (k) = (1 - eta(k))y(k-1) + eta(k)x(k). The authors first review and co mment on a technique of this form introduced by Schonauer and Weiss th at results in {y(k)) with monotone decreasing residual norms: this is referred to as minimal residual smoothing. Certain relationships betwe en the residuals and residual norms of the biconjugate gradient (BCG) and quasi-minimal residual (QMR) methods are then noted, from which it follows that QMR can be obtained from BCG by a technique of this form ; this technique is extended to generally applicable quasi-minimal res idual smoothing. The practical performance of these techniques is illu strated in a number of numerical experiments.