Inverse function, Taylor’s expansion, and high-order algorithms for n-th root computation

The computation of the n-th root r of a strictly positive real number r has a long history [1, 14]. In more recent work, for example in [10] and [20], continued fraction expansions are used to derive methods to compute r. Also, in [12] methods similar to those presented in [20] are obtained as special case of a determinantal family of root-finding methods [11]. For the computation of n-th root third order and fourth order methods are presented in [6]. General high order methods can be derived from the application of Newton’s method to an appropriate modified function [2] or using a modified Newton’s method applied to the original function [7, 8]. These two approaches have been revisited in [3]. Using combinations of basic functions identified for methods proposed in [2, 7], new high-order methods are derived for the computation of r in [15]. The goal of this paper is twofold. The first contribution is the presentation of a simple way to obtain two known families of high order methods to compute