Semilocal Convergence Behavior of Halley's Method Using Kantorovich's Majorants Principle
For researchers in numerical analysis, this work offers a refined theoretical framework for Halley's method, though it is an incremental improvement over existing majorant-based analyses.
The paper presents a new semilocal convergence analysis for Halley's method in Banach spaces, removing the need for a second root assumption in the majorizing function while maintaining Q-cubic convergence. It also provides a new error estimate and recovers Kantorovich and Smale-type results as special cases.
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.