Quotient sequence with symmetric signed quotients?

[paul zimmermann]

       Dear Niels,

> Is there any analysis of Euclid's algorithm, with different rounding of
> the quotients? E.g., for the fibonacci inputs gcd(21, 13), we have a
> standard quotient sequence of (1,1,1,1,1,1,1), taking 7 steps to get
> from (21,13) to (1,0).
> 
> If we instead try quotients rounded to nearest, we get
> 
> (21,13) ---->  (13, -5) ----> (-5, -2) ----> (-2, -1) ----> (-1, 0)
>         q = 2           q =-3          q = 2          q = 2
> 
> with 4 quotients rather than 7, and different from the standard continued
> fraction expansion.

the right person to ask is Brigitte Vallée:

https://vallee.users.greyc.fr/Publications/index.html

(Look for "nearest" or "odd" on this page.)

Paul