A robust Kantorovich's theorem on 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 Q-linearly to a zero of the non-linear operator under consideration. Using this result we show that 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 Newton method. 
O. P. Ferreira and B. F. Svaiter
math.NA