About Euclidean algorithm generalization

Hello,

I'm just subscribing and I don't have had yet time to read messages. I
would like only to say that I found yesterday something new, for me, about
mediant generalization question. I quote first from

<http://mathworld.wolfram.com/> :

<< Although attempts were made to generalize the >>
[ Euclidean algorithm to n >= 3 ]
<< by Hermite (1850), Jacobi (1868), Poincaré (1884), Perron (1907),
Brun (1919, 1920, 1957), and Szekeres (1970), all such routines
were known to fail in certain cases (Ferguson and Forcade 1979,
Forcade 1981, Hastad et al. 1989). The first successful integer
relation algorithm was developed by Ferguson and Forcade (1979)
(Ferguson and Bailey 1992, Ferguson et al. 1999). >>

Has someone here an idea on this last algorithm nature ?

If someone is interested to investigate in this way, I would ask also these
questions. Is it possible to build generalized Euclidean trees to represent
the Ferguson et al. algorithm like Euclid tree represents Euclidean
algorithm ? If so, although they won't be binary trees, maybe an enlarged
concept of transposed tree could be used, like Stern-Brocot tree is the
transposed tree of Euclid. If so, what are properties of such trees, and
does exist a simple way to build these presumable trees, like mediant
property allows to build Stern-Brocot tree ?

Pierre Lamothe

--- In harmonic_entropy@egroups.com, Pierre Lamothe <plamothe@a...>
wrote:
>
> Hello,
>
> I'm just subscribing and I don't have had yet time to read
messages. I
> would like only to say that I found yesterday something new, for
me, about
> mediant generalization question. I quote first from
>
> <http://mathworld.wolfram.com/> :
>
> << Although attempts were made to generalize the >>
> [ Euclidean algorithm to n >= 3 ]
> << by Hermite (1850), Jacobi (1868), Poincaré (1884), Perron
(1907),
> Brun (1919, 1920, 1957), and Szekeres (1970), all such routines
> were known to fail in certain cases (Ferguson and Forcade 1979,
> Forcade 1981, Hastad et al. 1989). The first successful integer
> relation algorithm was developed by Ferguson and Forcade (1979)
> (Ferguson and Bailey 1992, Ferguson et al. 1999). >>
>
> Has someone here an idea on this last algorithm nature ?

Thanks so much, Pierre, for finding this. It appears that the
mathematical developments we need may have just arisen at just the
right time . . .