An improved convergence theorem for the Newton method under relaxed continuity assumptions

TL;DR

This paper presents an improved convergence theorem for the Newton method that relaxes continuity assumptions on the derivative, using advanced majorization techniques to achieve better bounds and wider applicability.

Contribution

It introduces a novel 'first integral' approximation within the majorization framework, weakening convergence conditions and broadening initial guess options compared to prior results.

Findings

Weaker convergence conditions than previous theorems.

Enhanced bounds on solution location for F(x)=0.

Explicit restrictions for Lipschitz continuous derivatives.

Abstract

In the framework of the majorization technique, an improved condition is proposed for the semilocal convergence of the Newton method under the mild assumption that the derivative of the involved operator F(x) is continuous. Our starting point is the Argyros representation of the optimal upper bound for the distance between the adjacent members of the Newton sequence. The major novel element of our proposal is the optimally reconstructed 'first integral' approximation to the recurrence relation defining the scalar majorizing sequence. Compared to the previous results of Argyros, it enables one to obtain a weaker convergence condition that leads to a better bound on the location of the solution of the equation F(x)=0 and allows for a wider choice of initial guesses. In the simplest case of the Lipschitz continuous derivative operator, an explicit restriction is found which guarantees that the new convergence condition improves the famous Kantorovich condition.