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

In this paper, we study the explicit superlinear convergence rate of quasi-Newton methods. We particularly focus on the classical Broyden's methods for solving nonlinear equations and establish their explicit (local) superlinear convergence rates when the initial point is close enough to a solution and the approximate Jacobian is close enough to the exact Jacobian related to the solution. Our results provide the explicit superlinear convergence rates of Broyden's "good" and "bad" methods for the first time. The explicit convergence rates provide some important insights on the performance difference between the "good" and "bad" methods. The theoretical findings in the convergence analysis of Broyden's methods are also validated empirically in this paper.