  In this paper, we study the explicit superlinear convergence rates of
quasi-Newton methods. We particularly focus on the classical Broyden's method
for solving nonlinear equations. We establish its explicit (local) superlinear
convergence rate when the initial point is close enough to a solution and the
initial Jacobian approximation is also close enough to the exact Jacobian
related to the solution. Our results present the explicit superlinear
convergence rates of Broyden's "good" and "bad" update schemes. These explicit
convergence rates in turn provide some important insights on the performance
difference between the "good" and "bad" schemes, which are also validated
empirically.
