Andreas Enge on Wed, 15 Feb 2012 17:41:31 +0100

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Re: Complex AGM

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

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

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.