Residual Smoothing Techniques for Iterative Methods

Abstract

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.