  We present a new theoretical analysis of local superlinear convergence of
classical quasi-Newton methods from the convex Broyden class. As a result, we
obtain a significant improvement in the currently known estimates of the
convergence rates for these methods. In particular, we show that the
corresponding rate of the Broyden-Fletcher-Goldfarb-Shanno method depends only
on the product of the dimensionality of the problem and the logarithm of its
condition number.
