Implicit extra-update multi-step quasi-Newton methods

A new approach to the implicit update multi-step methods, introduced earlier by Ford and Tharmlikit (2003) and Moghrabi (2009), is addressed in this paper. Two algorithms are developed. One of the algorithms proposes a method that uses exact quantities in doing the update that are computed cheaply rather than relying on crude approximations, as is the case in the work of Ford (2001), Ford and Tharmlikit (2003) and Moghrabi (2009). The other method introduced here exploits El-Baali's (1999) extra update technique and is motivated by the improvements achieved by the application of such methods to the minimisation of nonlinear unconstrained problems. The implicit extra updates idea performs, on iterative basis, implicit updates to the Hessian approximation multiple times at each iteration. The updates are intended to be performed cheaply prior to updating the actual Hessian approximation at each iteration. The implicitly updated matrix serves as a metric for measuring distances among recent iterates, as required in the multistep Methods developed by Ford and Moghrabi (1993, 1994). The method displays numerical merits and motivates promising further investigation.