Explicit Superlinear Convergence of Broydens Method in Nonlinear Equations

In this paper, we follow the recent works about the explicit superlinear convergence rate of quasi-Newton methods. We focus on classical Broydens methods for solving nonlinear equations and establish explicit (local) superlinear convergence if the initial parameter and approximate Jacobian is close enough to the solution. Our results show two natural trade-offs. The first one is between the superlinear convergence rate and the radius of the neighborhood at initialization. The second one is the balance of the initial distance with the solution and its Jacobian. Moreover, our analysis covers two original Broydens methods: Broydens good and bad methods. We discover the difference between them in the scope of local convergence region and the condition number dependence.