RESIDUAL SMOOTHING TECHNIQUES FOR ITERATIVE METHODS

An iterative method for solving a linear system Ax = b produces iterates {x(k)} with associated residual norms that, in general, need not decrease ''smoothly'' to zero. ''Residual smoothing'' techniques are considered 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 comment on a technique of this form introduced by Schonauer and Weiss that results in {y(k)) with monotone decreasing residual norms: this is referred to as minimal residual smoothing. Certain relationships between 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 residual smoothing. The practical performance of these techniques is illustrated in a number of numerical experiments.