Accuracy of Two Three-term and Three Two-term Recurrences for Krylov Space Solvers

It has been widely observed that Krylov space solvers based on two three term re currences can give signi cantly less accurate residuals than mathematically equivalent solvers implemented with three two term recurrences In this paper we attempt to justify this di erence theoretically by analyzing the gap between the recursively and the explicitly computed residuals It is shown that in contrast with the two term re currences analyzed by Greenbaum SIAM J Matrix Anal Appl pp in the two three term recurrences the local roundo error contributions to the analyzed gap may dramatically amplify while propagating through the algorithm For the conjugate gradient method such a devastating behavior is however not observed frequently in practical computations where the di erence between three two term and two three term implementations is usually moderate or small This can be explained by our results We emphasize that in general there is no inherent weakness in the three term recurrence for the residual the di culty occurs when the iterate is also computed via a three term recurrence