the history of "iterations" begins certainly with the history of numerical
methods for solving a nonlinear equation.
For example, I've read in a French book that Newton has not explicitly proposed
such an iteration with the method that is now called "Newton-Raphson-Simpson
method ".
Newton (circa 1669) and Raphson (circa 1690) has proposed the algorithm/method
for polynomial equations with rational coefficients, Simpson has proposed the
algortihm for irrational and transcendental equations (circa 1740).
The book has been translated in English (it has a lot of original texts):
J.-L. Chabert "A history of algorithm" Springer 1998, ISBN 3-540-63369-3.
You will find also classical references about history of False Position methods
and Square roots computations.
Another very interesting book is the following:
R. Abraham, L. Gardini and C. Mira, "Chaos in discrete dynamical systems",
Springer-Telos, ISBN 0-387-94300-5.
The book has a lot of historical references (19th and 20th centuries essentially)
about "iterated maps". For example, they call the classical graph of
x[1] = f(x[0])
x[2] = f(x[1])
x[3] = f(x[2])
.
constructed with the points (x[i],f(x[i]) as the Koenigs-Lemeray method, it's
called the cobweb construction or the straircase method in English.
Pr. A. Douady has said to me some months ago that A. Caylay has writen a paper
about the Newton method and the equation x^3-1=0, the study of "basins of
attraction" of the method is now a classic in Chaos and Fractal theories but I've
no precise reference, Pr Douady didn't remember the exact references. If someone
has the hostorical reference....
Amicalement,
Robert Erra.
--E.S.I.E.A. 9 rue Vesale 75005 PARIS FRANCE.
James C. Taylor a ecrit:
> Can anyone help me place the first historic use(s) of iterations of the form: > (read x[0] as x sub 0) > > x[1] = f(x[0]) > x[2] = f(x[1]) > x[3] = f(x[2]) > . > . > . > > The high school at which I teach is performing Stoppard's "Arcadia", and > there is some discussion of this process and its place in the history of > mathematics. I would be interested not only in dates, but in people, > context(s), and references. > > Thanks! > James > > ******************************************************** > James C. Taylor > Computer Science and Information Technologies Department > Santa Fe Preparatory School > 1101 Camino de la Cruz Blanca > Santa Fe, NM 87501 > (505) 982-1829 x287 > (505) 982-2897 FAX > ********************************************************