Convergence rates for inexact Newton-like methods at singular points and applications

摘要

Newton's method converges linearly rather than quadratically if the initial guess is close to the solution and the Fréchet derivative is singular at the solution. Several authors have already tried to improve the rate of convergence for this method in this case. The limitations of their approach is twofold. First, the residuals at each step are assumed to be zero, which is not the case in general. Second, the previous works are limited only to the ordinary Newton method. Here we extend and improve on these results by providing convergence rates for inexact Newton-like methods on a Banach space setting. Moreover our results reduce to the ones already in the literature for special choices of the operators involved. We complete our study with some numerical examples appearing in connection with some “predator–prey” problems appearing in population modeling that show how our results improve on the earlier ones.