Explicit Superlinear Convergence Rates of Broyden's Methods in Nonlinear Equations

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.