Semilocal Convergence Behavior of Halley's Method Using Kantorovich's Majorants Principle

TL;DR

This paper analyzes the semilocal convergence of Halley's method in Banach spaces using Kantorovich's majorants, establishing Q-cubic convergence without requiring a second root of the majorant, and providing new error estimates.

Contribution

It introduces a novel convergence analysis for Halley's method that relaxes previous assumptions and extends convergence results based on Kantorovich's and Smale's conditions.

Findings

Guarantees Q-cubic convergence without second root assumption

Provides new error estimates based on directional derivatives

Includes special cases for Kantorovich and Smale type conditions

Abstract

The present paper is concerned with the semilocal convergence problems of Halley's method for solving nonlinear operator equation in Banach space. Under some so-called majorant conditions, a new semilocal convergence analysis for Halley's method is presented. This analysis enables us to drop out the assumption of existence of a second root for the majorizing function, but still guarantee Q-cubic convergence rate. Moreover, a new error estimate based on a directional derivative of the twice derivative of the majorizing function is also obtained. This analysis also allows us to obtain two important special cases about the convergence results based on the premises of Kantorovich and Smale types.