A quasi-Newton method with optimal r-order without independence assumption∗

Quasi-Newton methods based on least change secant updating formulas that solve linear equations Ax = b in n = dim(x) = dim(b) steps can be expected to solve corresponding smooth nonlinear systems n-step quadratically, i.e. with an r-order of ρ = 2 = 1 + 1/n+O(1/n). The best rate one can possibly expect on general problems is given by the positive root ρn of ρ (ρ− 1) = 1, for which ρn − 1 = ln(n)/n+O(1/n). To show that this upper bound is actually achieved one usually has to impose a priori some kind of linear independence condition on the sequence of steps taken by the quasi-Newton iteration in question. Without any such assumptions we establish in this paper the convergence order ρn for the two-sided rank one formula proposed by Schlenkrich et al in [SGW06]. It requires the evaluation of adjoint vectors, is invariant with respect to linear transformations on the variable domain and combines the properties of bounded deterioration and heredity.