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 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. Andreas