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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.