Abstract
The classical quadratically convergent Newton-Raphson iterative scheme for successive approximations of a root of an equation $f(t)=0$ has been extended in various ways by different authors, going from cubical convergence to convergence of arbitrary orders. We introduce two such extensions, using appropriate differential operators as well as combinatorial arguments. We conclude with some applications including special series expansions for functions of the root and enumeration of classes of tree-like structures according to their number of leaves. Le schéma itératif classique à convergence quadratique de Newton-Raphson pour engendrer des approximations successives d'une racine d'une équation $f(t)=0$ a été étendu de plusieurs façons par divers auteurs, allant de la convergence cubique à des convergences d'ordres arbitraires. Nous introduisons deux telles extensions en utilisant des opérateurs différentiels appropriés ainsi que des arguments combinatoires. Nous terminons avec quelques applications incluant des développements en séries exprimant des fonctions de la racine et l'énumération de classes de structures arborescentes selon leur nombre de feuilles.
Highlights
Letn≥0 be a sequence of real numbers converging to a
The convergence is said to be of order p if tn+1 − a = O ((tn − a)p), as n → ∞. This means that the convergence is very rapid, when p ≥ 2, since the number of correct decimal digits in the approximation of a is essentially multiplied by p at each step
It is interesting to note that the function defined by the right members of (20-21) is a constant function in a neigborhood of a
Summary
Let (tn)n≥0 be a sequence of real numbers converging to a. Cubical convergence (p = 3), can be achieved using Householder’s method (see Householder (1970)), f (t). To achieve convergence of order k + 1, k ≥ 3, one can use the general Householder’s method (see Householder (1970)),. Another way to achieve arbitrary order convergence is to make use of the method of indeterminate coefficients together with a Taylor expansion around the root (see Sebah and Gourdon (2001)).
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