Re: How to determine Mod(a,b) with t_COMPLEX b?

* [email protected] [2025-05-27 00:01]:
[...]
> The minimal residue of 1+4*I modulo 3+2*I is the yellow point -I in the
> example:
> https://en.wikipedia.org/wiki/Gaussian_integer#Describing_residue_classes
> 
> How can minimal residue of an input gaussian integer modulo a gaussian
> integer be computed in PARI/GP?

? a = 1+4*I; b = 3+2*I;
? a - round(a/b)*b
%2 = -I

This is not exacly the same normalization as in the Wikipedia article,
because ties are rounded up (= floor(x+1/2)), not down (= ceil(x-1/2)),
but it has the same properties (defines a Euclidean division with unique
quotient and remainder).

If you insist on the same (awkward) normalization, then you must use
something like

  myround(z) = ceil(real(z)-1/2) + I * ceil(imag(z)-1/2);
  a - myround(a/b)*b

instead.

? round(1/2 + I/2)
%3 = 1 + I
? myround(1/2 + I/2)
%4 = 0

Cheers,

    K.B.
-- 
Pr. Karim Belabas, U. Bordeaux, Vice-président en charge du Numérique
Institut de Mathématiques de Bordeaux UMR 5251 - (+33) 05 40 00 29 77
http://www.math.u-bordeaux.fr/~kbelabas/