A family of Chebyshev-Halley type methods in Banach spaces

Abstract

A family of third-order iterative processes (that includes Chebyshev and Halley's methods) is studied in Banach spaces. Results on convergence and uniqueness of solution are given, as well as error estimates. This study allows us to compare the most famous third-order iterative processes.

Graphic and numerical comparison between iterative methods

“…When β → −∞, we get Newton's method. This family was studied by Werner in 1980 (see [19]), and can also be found in [3, p. 219] and [10]. It is interesting to note that any iterative process given by the expression…”

Generalized Newton Raphsons method free from second derivative

“…This is so-called the Halley's method [6,11,12] for root-finding of nonlinear functions, which converges cubically. Simplification of (2.2) yields another iterative method as follows:…”

“…But the reality is that some of the higher order iterative methods have vast applications and have best performance as compared to those which have low order of convergence. No doubt, higher order iterative methods require more functional evaluations which is the main drawback of these methods [11,16,24]. We are interested in finding higher order iterative method free from second derivative.…”

Certain improvements of Newton’s method with fourth-order convergence

“…We used the test functions in Weerakoon and Fernando [2] and in Neta [12] We present some numerical test results for various cubically convergent iterative schemes in Table 1. Compared were Newton method(NM), the method of Weerakoon and Fernando (WF) defined by (3), Halley's method [24,25] (HalleyM) defined by…”