A robust Kantorovich’s theorem on the inexact Newton method with relative residual error tolerance

We prove that under semi-local assumptions, the inexact Newton method with a fixed   relative residual error tolerance converges QQ-linearly to a zero of the nonlinear operator under consideration. Using this result we show that the Newton method for minimizing a self-concordant function or to find a zero of an analytic function can be implemented with a fixed relative residual error tolerance.In the absence of errors, our analysis retrieve the classical Kantorovich Theorem on the Newton method.