Symmetric-rank-one multi-step quasi-Newton implicit update algorithms

Implicit multi-step quasi-Newton methods, introduced in [1], use the existing Hessian approximation to compute, at each iteration, the parameters required in the interpolation. To avoid the burden of computing the needed matrix-vector products, required by this approach, approximations based on the Secant Equation were proposed. Based on [2], a different approach to dealing with this difficulty was suggested, in which standard single-step quasi-Newton updates were replaced by successive iterations, by two-step updates, so that approximations were no longer necessary. The recent research has shown that the quantities required to compute the parameters referred to the above may be exactly computed by means of recurrence, so that the technique of alternation is no longer the only alternative. In this paper, we consider the derivation of new recurrences for the implicit update methods based on the well-known Symmetric Rank One (SRI) update formula. We present the results of a range of numerical experiments to compare and evaluate the methods developed here.