A robust Kantorovich's theorem on the inexact Newton method with relative residual error tolerance (Q423882)

The authors state some properties of the majorant function, and establish the relationship between the majorant function and the nonlinear operator in solving nonlinear equations. A family of regions where the behavior of the inexact Newton iteration is estimated using the majorant function is introduced. The union of all those regions is shown to be invariant under the inexact Newton iteration with a fixed relative residual error tolerance. A convergence analysis of the inexact Newton method with relative error is presented. The authors show that the Newton method for finding a zero of an analytic function under the usual semi-local assumption of the \(\alpha\)-theory can be implemented with a fixed relative residual error tolerance.