Residual smoothing techniques: do they improve the limiting accuracy of iterative solvers?

Abstract

Many iterative methods for solving linear systems, in particular the biconjugate gradient (BICG) method and its "squared" version CGS(orBICGS), produce often residuals whose norms decrease far from monotonously, but fluctuate rather strongly. Large intermediate residuals are known to reduce the ultimately attainable accuracy of the method, unless special measures are taken to counteract this effect. One measure that has been suggested is residual smoothing: by application of simple recurrences, the iterates $x_n$ and the corresponding residuals $r_n$ :≡ $b - Ax_n$ are replaced by smoothed iterates $s_n :≡ b - Ay_n$. We address the question whether the smoothed residuals can ultimately become markedly smaller than the primary ones. To investigate this, we present a roundoff error analysis of the smoothing algorithms. It shows that the ultimately attainable accuracy of the smoothed iterates, measured in the norm of the corresponding residuals, is, in general, not higher than that of the primary iterates. Nevertheless, smoothing can be used to produce certain residuals, most notably those of the minimum residual method, with higher attainable accuracy than by other frequently used algorithms.