> >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])
> > .
> > .
> > .
>
> See Ptolemy's _Almagest_ (X 7) for his fixed-point iteration for
> determining the eccentricities of the deferents of the superior planets
> (O. Neugebauer's _History of Ancient Mathematical Astronomy_, I pp. 174--178,
> gives a helpful exposition).The method probably considerably predates
> Ptolemy, but it may be difficult to find an explicit discussion of it in
> the classic Hellenic mathematical works, since it falls far short of
> Euclidean standards of rigorous demonstration.It appears in Indian
> texts from the first few centuries of this era, but not identifiably
> earlier than the Almagest, as far as I know; I don't know what evidence
> there may be for such a method in Egyptian or Chinese math.Good luck,
>
> Kim Plofker
> Dept. of History of Mathematics
> Brown University
>
According to Boyer's _A History of Mathematics_ the ancient mesopotamians
applied iterative procedure for extracting square roots. If x is a
first approximation to the root of a, then a better approximation is
1/2(x + a/x), etc. He also states that the method is sometimes attributed
to "later men", such as Archytas (428-365 BC) or Heron (ca 100), or Newton.
Avinoam Mann