Newtonʹs method and generation of a determinantal family of iteration functions

It is well known that Halleyʹs method can be obtained by applying Newtonʹs method to the function f/f′. Gerlach (SIAM Rev. 36 (1994) 272–276) gives a generalization of this approach, and for each m⩾2, recursively defines an iteration function Gm(x) having order m. Kalantari et al. (J. Comput. Appl. Math. 80 (1997) 209–226) and Kalantari (Technical Report DCS-TR 328, Department of Computer Science, Rutgers University, New Brunswick, NJ, 1997) 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, this gives a new insight on the Basic Family and Newtonʹs method.