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On the semilocal convergence behavior for Halley’s method. (English) Zbl 06348494
Summary: 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.

##### MSC:
 47J05 Equations involving nonlinear operators (general) 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 65H10 Numerical computation of solutions to systems of equations
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