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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 OukAkkxk. Building on earlier ideas on residual replacement a n d on insights in the nite precision behaviour of the Krylov subspace methods, computable error bounds are derived for iterations that involve occasionally replacing the computed residuals by the true residuals, and theydoi:10.1137/s1064827599353865 fatcat:3mntql6kz5baxdtxt6hisw43ly