The use of alternation and recurrences in two-step quasi-Newton methods

Recent work has shown that the quantities required to compute the parameters referred to above may be computed exactly by means of a recurrence, so that the technique of alternation is no longer the only alternative if we wish to avoid approximations. In this paper, we describe the derivation of this recurrence. We present the results of a range of numerical experiments to compare and evaluate the three approaches of approximation, alternation, and recurrence. Finally, we show how the use of recurrences may be extended to multistep methods employing three or more steps.