Sharp estimation of local convergence radius for the Picard iteration

We investigate the local convergence radius of a general Picard iteration in the frame of a real Hilbert space. We propose a new algorithm to estimate the local convergence radius. Numerical experiments show that the proposed procedure gives sharp estimation (i.e., close to or even identical with the best one) for several well known or recent iterative methods and for various nonlinear mappings. Particularly, we applied the proposed algorithm for classical Newton method, for multi-step Newton method (in particular for third-order Potra-Ptak method) and for fifth-order "M5" method. We present also a new formula to estimate the local convergence radius for multi-step Newton method.