Re: Motion 11
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Dear Dan,
Dan Zuras Intervals wrote:
> I believe your observation about Corollary 1 & arguments
> for the need for reverse operations both contribute to an
> approach along the lines George has suggested.
>
> For example, Nate has pointed out the the lack of inverse
> operators within ordinary intervals can be repaired by the
> use of the Dual() operator. Where
>
> Dual([a,b]) = [b,a].
>
> Thus, you can solve for yy in xx + yy = zz by
>
> xx + yy = zz
> (xx - Dual(xx)) + yy = zz - Dual(xx)
> [0,0] + yy = zz - Dual(xx)
> yy = zz - Dual(xx)
>
> All 4 basic operations may be inverted in this way. (I
> will leave it to Nate to discuss the details on grounds
> of incompetence on my part. :-)
I do not know enough about modal interval arithmetic to have any
definitive argument about that. However, I doubt you can easily obtain
the same results as with reverse division. Once again, consider my example:
A=[0,2]
B=[-1,1]
C=[1,1]
with \circ=\times
and compute \times_1^-(B,C,A) = hull({x\in[0,2]\mid\exists b\in[-1,1],
x\times b\in[1,1]})
Can you obtain [1, 2] as a result? The n+1-ary reverse division will.
> Further, it is my understanding that inverse operations
> (for example, in Newton steps) can lead to tighter
> intervals than those defined in Motion 11. Of course,
> one must intersect it with the original guess interval
> just as in Motion 11.
I am lost here. What do you call inverse operations?
> So I guess I'm suggesting the addition of the Dual()
> operator as something out of which the needed operators
> can be constructed.
>
> Does this work?
> Would this be a simpler approach along the lines George
> has mentioned?
> Is anything else needed?
As I said, there is more to reverse operations than division. How would
reverse cosine be implemented with Dual (see example of use below)?
> In particular, are the 3-op forms needed given that we
> can just intersect the 2-op form with the original guess?
>
> Anyone? - Dan
No, because Corollary 1 is false.
You just have to consider the example above. Another one, taken from the
first paper I referred you to in a previous mail:
Say you know that x\in[20,26], and you have the *relation*: y=cos x,
with y\in[-0.3,0.2].
What can you infer about x from that? If you use some unary reverse
cosine, you cannot obtain anything useful for x. You have to use a
binary reverse cosine to take into account x's original domain *inside*
the operator. You will then be able to infer that x\in[6\pi+acos 0.2,
8\pi-acos 0.2]\approx[20.22,23.77].
F.
- --
Frédéric Goualard LINA - UMR CNRS 6241
Tel.: +33 2 51 12 58 38 Univ. of Nantes - Ecole des Mines de Nantes
Fax.: +33 2 51 12 58 12 2, rue de la Houssinière - BP 92208
http://goualard.frederic.free.fr/ F-44322 NANTES CEDEX 3
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