Re: Comments on Motion 27-A "Decorated Intervals"
Jürgen, Marco, Vincent, P1788
On 15 Jul 2011, at 23:32, Vincent Lefevre wrote:
>>> About Remark 5 (and the whole motion), what do the decorations really
>>> mean in practice? For instance, consider f(x) = tan(floor(x)*pi+pi/2),
>>> implemented as the composition of the various basic functions, and its
>>> interval extension. At Level 1, it is ndf for X = [1,1.5], but not at
>>> Level 2, where it is con (because of rounding). With X = [1,3], it is
>>> con at both Level 1 and Level 2, though f(x) is nowhere defined. So,
>>> what confidence con brings here?
>>
>> The decoration con tells you that that the level 2 computation of the
>> function f(x) may have a singularity on the interval x.
>
> I meant: what confidence con brings compared to ndf?
>
> Since
> * one can obtain con for a function that is nowhere defined, and
> * one can obtain ndf for a function that is not nowhere defined
> (due to the tracking rules),
> it seems that there isn't a difference between ndf and con.
From my reading of the text, I agree. It's not the _mechanism_ of propagating decorations that is at fault -- that (Def 7, Def 8) is the one I think we are all agreed on.
What's wrong is Theorem 3. If that is all one can trust to be true, it seems to me the current draft can't handle Nate's branch and bound application. Suppose one computes ndf. According to Theorem 3, con might be true, or even saf! But Nate's B&B relies on knowing that if you compute ndf then f is definitely nowhere defined on xx.
Arnold and I expect to have a revised decoration document out tomorrow. The Fundamental Theorem is hopefully much KISSed. We are now confident that its proof holds water, including when the input box has empty components.
John