Theoretical and numerical aspects of a non-stationary preconditioned iterative method for linear discrete ill-posed problems

This work considers some theoretical and computational aspects of the recent paper (Buccini et al., 2021), whose aim was to relax the convergence conditions in a previous work by Donatelli and Hanke, and thereby make the iterative method discussed in the latter work applicable to a larger class of problems. This aim was achieved in the sense that the iterative method presented convergences for a larger class of problems. However, while the analysis presented is correct, it does not establish the superior behavior of the iterative method described. The present note describes a slight modification of the analysis that establishes the superiority of the iterative method. The new analysis allows to discuss the behavior of the algorithm when varying the involved parameters, which is also useful for their empirical estimation.