TY - GEN A1 - Griewank, Andreas A1 - Schlenkrich, Sebastian A1 - Walther, Andrea T1 - A quasi-Newton method with optimal R-order without independence assumption N2 - 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 $\rho = 2^{1/n} = 1 + 1/n +O(1/n^2)$. The best rate one can possibly expect on general problems is given by the positive root $\rho_n$ of $\rho^n(\rho -1)=1$, for which $\rho_n-1 = \ln(n)/n + O(1/n^2)$. 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 $\rho_n$ for the two-sided rank one formula proposed by Schlenkrich et al in \cite{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. KW - nonlinear equations KW - quasi-Newton methods KW - convergence order KW - adjoint based updating Y1 - 2006 UR - https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/351 UR - https://nbn-resolving.org/urn:nbn:de:0296-matheon-3514 ER -