Newton's method under mild differentiability conditions with error analysis

Abstract

The concept of majorizing sequences introduced by Rheinboldt (SIAM J.N.A. 1968) is used to prove convergence for Newton's method for operator equations of the formT f=? when the operator satisfied the condition that the Frechet derivative is Holder continuous. A detailed analysis of computational errors is given for Newton's method applied to operators with Holder continuous derivatives. This analysis is shown to reduce the analysis of Lancaster (Num. Math. 1968) when the operator has a continuous second derivative. The above analysis is applied to an example of a second order differential equation.