Karim Belabas on Sat, 10 Mar 2007 19:14:00 +0100

 Re: Desired behaviour ?

```* Loic Grenie [2007-03-10 17:41]:
>     I'm just wondering if there is a strong reason why 1.*I*I has an
>   imaginary part (equal to 0., but it's there). For instance neither I*I*1.
>   nor 1.*(I*I) have any imaginary part.

1) Remember that floating point arithmetic is not associative, and need not
be commutative.

2) Using the standard formulas for complex arithmetic

(a+bI)(c+dI) = (a*c - b*d) + I (a*d + b*c)

[ 4 multiplications ], we obtain:

(0 + 1.*I) * (0 + I) = (0*0 - 1.) + I*(0*1. + 0*1) = -1.

and imaginary part is indeed an exact 0.

3) Unfortunately we use Karatsuba (aka 3M) instead of standard formulas,
and compute

(a+bI)(c+dI) = (a*c - b*d) + I ((a+b)*(c+d) - a*c - b*d)

[ 3 multiplications ]. There the imaginary part becomes

(0 + 1.)*(0 + 1) - (0*0) - (1.*1) = 0.,

NOT an exact 0 !

So, yes, this is expected.

Cheers,

K.B.
--
Karim Belabas                  Tel: (+33) (0)5 40 00 26 17
Universite Bordeaux 1          Fax: (+33) (0)5 40 00 69 50
351, cours de la Liberation    http://www.math.u-bordeaux.fr/~belabas/
F-33405 Talence (France)       http://pari.math.u-bordeaux.fr/  [PARI/GP]

```