Bill Allombert on Thu, 16 Feb 2012 22:02:45 +0100

 Re: Complex AGM

```On Wed, Feb 15, 2012 at 05:41:24PM +0100, Andreas Enge wrote:
> On Wed, Feb 15, 2012 at 04:20:45PM +0000, John Cremona wrote:
> > Yes, that is essentially Cox's definition.  But this ambiguous case
> > only happens at the first step of the algorithm anyway, and when it
> > does happen the two limits you get by making both choices have exactly
> > the same absolute value.
>
> And they are mirror images with respect to the axis given by the two input
> values.
>
> After discussion with Bill, we have a better suggestion. If in the first step
> the choice is ambiguous, choose the one that yields an angle of pi/2 between
> the arithmetic and the geometric mean, and not of 3pi/2. If I am not mistaken,
> this definition makes the AGM completely homogeneous:
> AGM (omega*a, omega*b) = omega * AGM (a, b), even if a/b is a negative real
> number.
>
> This gives a nice intrinsic definition (actually, two equally valid ones,
> since one could have chosen 3pi/2 over pi/2; so maybe one should say a
> coherent definition). In particular, we can still use the current implemen-
> tation that normalises one entry to 1. Or more precisely, there is a choice
> to make in the first step of normalising towards 1 or -1 so that afterwards,
> the canonical choice of square root with positive imaginary part corresponds
> to the AGM defined as above.

I have commited that change to agm. Thanks for all the comments!
Now I can move to ellpointoz.

Cheers,
Bill

```