This paper presents generalizations of the two results which have been useful for analyzing methods of the form $x_{k+1} = x_{k} - B_{k}^{-1}F(x_{k})$. The bounded deterioration theorem of Broyden-Dennis-More is generalized to show that if {$B_{k}$} or {$B_{k}^{-1}$} is of bounded deterioration as a sequence of approximants to some $B_{}$ or $B_{}^{-1}$ then the iteration above has the same local convergence properties and arbitrarily nearly the same linear rate as would be achieved by the stationary iteration function which uses $B_{k} = B_{}$. The characterization theorem for superlinear convergence given by Dennis-More is then generalized to give conditions under which the rates are the same. In the case when $B_{} = F'(x_{*})$, these results reduce to those already known.