Re: Motion 11 questions & comments...
> Date: Fri, 19 Feb 2010 17:59:35 +0100
> From: Dominique Lohez <dominique.lohez@xxxxxxx>
> To: Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>
> CC: Nate Hayes <nh@xxxxxxxxxxxxxxxxx>, stds-1788@xxxxxxxxxxxxxxxxx
> Subject: Re: Motion 11 questions & comments...
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], 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?
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.
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).
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,
Dan
>
> Dan Zuras Intervals a =E9crit :
> >> From: "Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>
> >> To: "P1788" <stds-1788@xxxxxxxxxxxxxxxxx>
> >> Subject: Re: Motion 11 questions & comments...
> >> Date: Mon, 8 Feb 2010 12:48:20 -0600
> >>
> >
> > Ah, thank you Nate. I see I was confused about the
> > nature of reverse operations altogether.
> >
> > Then, Marco, I am also confused as to the purpose
> > of these operations.
> >
> > . . .
> >
> > This business of returning a pair of intervals does
> > not generalize well beyond the simple pole in divide.
> >
> > Neither does it play well in expressions.
> >
> > I think my confusion has graduated to wariness & doubt.
> >
> > Can you allay my fears by explaining further?
> >
> >
> >
> > Dan
> >
> >
> Dan,
> I understand and share you fears and doubt.
> IMHO the troubles arise from a too ambitious purpose assigned to the
> reverse functions
> namely to define at conceptual level 1 a kind of equations solver.
>
> My own understanding of these functions is that they should be defined
> at level 2 without any level 1 precursor.
>
> For the shake of the discussion, I only consider the reverse function of
> a function f(x) of a single variable
>
> We are trying to find solutions of the interval equation F(X) = Y0
> where Y0 are interval in IF not IR
>
> Given Y0, we know a rough approximation of X namely A (of course a
> member of IF)
> This means that Y0 must be included into F(A)
> The reverse functions are tools that should allow us to refine the
> initial guess A.
> Assuming the A can be written A=[a,b] we search with the following refi=
> nment
>
> - Left refinement LR an interval [a,x] such that
> - Y0 is included into F(LR)
> - LR is included into any interval
> U=[a,u] such Y0 is included into F(U)
>
> - Similarly a right refinement RR an interval [y,b] such tha=
> t
> - Y0 is included into F(RR)
> - LR is included into any interval
> V=[v,b] such Y0 is included into F(V)
>
> -When LR and RR intersect each other the the best refinement
> BR can be written BR=[y,x]
>
> -When LR and RR do not intersect they are valid refinement.
> A further central refinement CR=[x,y] can be defined whenever
> Y0 is included into F(CR)
>
>
>
> . . .
>
> Defining a conceptual model on IR is of course possible. However it
> should not be carried since it is misleading.
> For example, due the the interval rounding , solution lying in CR at
> level 1 may be moved to LR at level 2.
> Best regards
>
> Dominique
>
> . . .
>
> --
> Dr Dominique LOHEZ
> ISEN
> 41, Bd Vauban
> F59046 LILLE
> France