Residual Smoothing Techniques for Iterative Methods

An iterative method for solving a linear system Ax b produces iterates {xk with associated residual norms that, in general, need not decrease "smoothly" to zero. "Residual smoothing" techniques are considered that generate a second sequence {Yk via a simple relation yk (1 0k)yk- + r/kxk. The authors firstreview andcomment on a technique of this form introduced by Sch6nauer and Weiss that results in {Yk} 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 ofthese techniques is illustrated in anumber of numerical experiments.