AbstractIt is well-known that Halley’s method can be obtained by applying Newton’s method to the function f/√ f ′ . Gerlach , gives a generalization of this approach, and for each m ≥ 2, recursively defines an iteration function Gm(x) having order m. Kalantari et al. , and Kalantari  derive and characterize a determinantal family of iteration functions, called the Basic Family, Bm(x), m ≥ 2. In this paper we prove, Gm (x) = Bm (x). On the one hand, this implies that Gm(x) enjoys the previously derived properties of Bm(x), i.e., the closed formula, its efficient computation, an expansion formula which gives its precise asymptotic constant, as well as its multipoint versions. On the other hand, this gives a new insight on the Basic Family and Newton’s method.
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