Variants of residual smoothing with a small residual gap

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Short-recurrence Krylov subspace methods, such as conjugate gradient squared-type methods, often exhibit large oscillations in the residual norms, leading to a large residual gap and a loss of attainable accuracy for the approximate solutions. Residual smoothing is useful for obtaining smooth convergence for the residual norms, but it has been shown that this does not improve the maximum attainable accuracy in most cases. In the present study, we reformulate the smoothing scheme from a novel perspective. The smoothed sequences do not usually affect the primary sequences in conventional smoothing schemes. In contrast, we design a variant of residual smoothing in which the primary and smoothed sequences influence each other. This approach enables us to avoid the propagation of large rounding errors, and results in a smaller residual gap, and thus a higher attainable accuracy. We present a rounding error analysis and numerical experiments to demonstrate the effectiveness of our proposed smoothing scheme.