Generalized Kantorovich-type theorem for the Fixed Slope Iterations

TL;DR

This paper generalizes the Kantorovich theorem for a modified Newton method using a fixed invertible operator, relaxing continuity assumptions and providing broader convergence guarantees and error bounds.

Contribution

It extends the Kantorovich theorem to a broader class of fixed slope iterations with weaker conditions and improved convergence and error estimates.

Findings

Generalized Kantorovich theorem for fixed slope iterations.

Broader convergence domain and weaker conditions.

Finer error bounds compared to previous results.

Abstract

The extended modification of the Newton method is considered when the inverse of the derivative (of the operator F(x) in the equation F(x)=0) is replaced by an invertible bounded x-independent operator B. The continuity assumption is relaxed to the requirement that F(x) is continuously Frechet-differentiable. The Kantorovich majorization technique is adapted to formulate and prove the corresponding generalization of the Kantorovich theorem originally stated for the standard modified Newton method (MNM) when the derivative is Lipschitz continuous. In the MNM case, the generalized theorem is shown to extend the existing one due to Argyros. For a generic B and a Holder continuous derivative, the proposed theorem leads to a weaker condition of the semilocal convergence, larger uniqueness domain and finer error bounds compared to the previous results of Ahues and Argyros.