Re: Motion 11 questions & comments...
Dan Zuras Intervals a écrit :
Dan,
Dominique,
I believe Marco is correct to try to define these things
at level 1. We may disagree on the details but I believe
he is in the right neighborhood.
Further, your level 2 definition of such an inverse
function seems clever but it is a bit too subtle for me
to figure out on first reading. You say that it does
not depend on the derivative but in looking for a small
interval that surrounds each endpoint you seem to depend
on the monotonicity of the function in question.
Although I may be misinterpreting that.
Still, to make your example concrete, for f(x) = x^2 in
the equation f(xx) = [1,2],
In addition we must start with an initial "guess" for xx which
correspond to
the interval used to calculate the intersection in Marco's formulation.
Let us assume the value GI=[0,3] for the guess
The left refinement LR is LR = [0, roundup(sqrt(2))]
The right refinement LR is LR = [round down(sqrt(1)), 3]
The intersection of these intervals does not vanishes so the best
refinement BR is meaningful
We get BR = [round down(sqrt(1)), roundup(sqrt(2))]
BR is a correct refinement of the initial guess.
you find the smallest left
interval [a,x] such that [1,2] is in f([a,x]) & the
smallest right interval [y,b] such that [1,2] is in
f([y,b]) but rather than return [a,b] you return [x,y].
This is the subtle point which eludes me. Are we not
looking for [a,b] to account for roundoff error in the
representation of the endpoints? In this case, are we
not looking for [1,roundup(sqrt(2))] which f() would
map slightly OUTSIDE of [1,2]?
In either case, are we not missing the interval segment
[-roundup(sqrt(2)),-1] which Marco is looking for?
It depends on the starting guess interval. But this also true in Marcoś
Formulation.
For the previous example, if we start with a guess interval GI=[-3, +3]
We can write
LR =[-3, roundup(-sqrt(1))]
RR=[round down(sqrt(1)), 3]
Since the intersection of LR and BR is now empty
Each interval has to dealt with as a possible refinement.
In contrast the central refinement CR= [roundup(-sqrt(1)), round
down(sqrt(1))] has not to be considered further since
F(CR) does not include [1, 2]
Applying the same process we can
refine LR to LR =[-roundup(sqrt(2)), roundup(-sqrt(1))].
Similarly RR can be refined to [round down(sqrt(1)), roundup(sqrt(2))}
At these points, no further refinement can be found.
Still, I am intrigued by your approach. Can one define
(at level 1) a series of ever narrowing xx(i) such that
xx(i) is contained in xx(j) for all i>j and f(xx(i))
contains Y0 for all i? The last xx(i) being our result,
of course.
Yes. This is shown in the second example.
Again, we would have problems with monotonicity &
multiple interval segments that solve f(X) = Y0 but
I suspect those can be dealt with by a suitable choice
of xx(0).
I may have missed some hidden hypothesis, but as as far as I know the
only hypothesis used may be written
If interval X is included into interval X' then F(X) is included into F(X')
Marco's issue of needing to select the proper interval
segment still applies. He does it by intersection
before the hull. Perhaps this way, it could be done
by proper choice of xx(0). Namely, the starting
interval Marco would intersect the result with.
Is there merit here?
Yours,
Dominique
--
Dr Dominique LOHEZ
ISEN
41, Bd Vauban
F59046 LILLE
France
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Email: Dominique.Lohez@xxxxxxx