“…Early work which uses the Broyden formula to update the preconditioner was introduced in [14]. Other updates based on matrices of small rank were considered in [15], [16], [17] and in limited-memory variable metric methods [18] for smooth optimization, to name just of few. An important class of algebraically-motivated strategies to accelerate the convergence of preconditioned iterative methods is based on constructing or improving the preconditioner by adaptive spectral information obtained directly from the Krylov subspace methods, see, e.g., [19,20,21,22,23,24,25,26].…”