G. H. Brown jun.
[Am. Math. Monthly 84, 726-728 (1977; Zbl 0375.65025
)] and G. Alefeld
[ibid. 88, 530-536 (1981; Zbl 0486.65035
)] applied Newton’s method to
being a real-valued function of a real variable, to solve
approximately, and thus obtained Halley’s method for
. In the present paper this idea is used in a generalized form. It turns out that, among others, the following statement is true. Let
. Then Halley’s formula with 1/2 replaced by
gives an iteration formula which converges of order at least
to the (simple) zero
(provided the starting term in the sequence is chosen sufficiently close to
). Some examples with numerically computed errors are given.