Residual Replacement Strategies for Krylov Subspace Iterative Methods for the Convergence of True Residuals

In this paper, a strategy is proposed for alternative computations of the residual vectors in Krylov subspace methods, which improves the agreement of the computed residuals and the true residuals to the level of O(u) ||A|| ||x||. Building on earlier ideas on residual replacement and on insights in the finite precision behavior of the Krylov subspace methods, computable error bounds are derived for iterations that involve occasionally replacing the computed residuals by the true residuals, and they are used to monitor the deviation of the two residuals and hence to select residual replacement steps, so that the recurrence relations for the computed residuals, which control the convergence of the method, are perturbed within safe bounds. Numerical examples are presented to demonstrate the effectiveness of this new residual replacement scheme.