RE: Another proposal for a "split" function to complement "mid"
Vladik Kreinovich wrote:
> If I understand correctly, we want to have approximately sqrt(a,b)
> when a and b are large, and approximately (a+b)2/ when a and b are
> small, but in this proposal, if a >> b, then (a+b)/2 >> sqrt(ab),
> hence we would get asymptotically a/4 instead of sqrt(ab).
I was sort of aware of this skewing effect, but Vladik is right: the
Arithmetic Mean continues to dominate, and one AGM step does not help
much.
> I think what Michel meant was geometric mean 2/(1/u + 1/v) = 2uv/(u+v),
> where u = (a+b)/2 and v =sqrt(ab). This will lead to approximately
> sqrt(ab) when a >> b and a and b are large and to (a+b)/2 when a >>b
> and a and b are small -- just as John said we want.
Great! So we combine all three types: HM(AM,GM)
The 2/(1/u+1/v) is the Harmonic Mean HM(u,v).
(v = GM(a,b) = sign(a)*sqrt(a*b) is the signed Geometric Mean.
We still need to treat intervals containing zero specially,
as the GM is not defined when a*b < 0.
I still owe John a resolution of the bunching problem for the
convert-to-sortable method. My partial AGM was a quick thought
along the way; I need to find time to think more deeply.
Michel.
---Sent: 2012-03-20 15:39:31 UTC