Re: Re-submission of motion 5: multiple-format arithmetic.
Arnold Neumaier wrote:
> Nate Hayes schrieb:
>> Arnold Neumaier wrote:
>>> Nate Hayes schrieb:
>>>
>>> Moore's theroem remains valid with the proposed definition, since
>>> a real expression defines a real function f(x_1, ..., x_n) only
>>> on domains where no divisor is zero.
>>>
>>> Therefore, correctly, for xx=[0,1]
>>> {1/(x^2-x+1) | x in xx}
>>> subseteq 1/(xx^2-xx+1) = 1/([0,1]-[0,1]+1) =
>>> 1/[0,2]=[1/2,inf], while with your definition, division by an
>>> interval containing zero gives NaN), and we'd get NaN, violating
>>> Moore's law since {1/(x^2-x+1) | x in xx} subseteq NaI
>>> does not hold.
>>
>> The reason it "violates Moore's law" is because x = 0 is not in the
>> domain of the function! That's the whole point of returning NaI...
>
> ???
>
> When I evaluate f(x):=1/(x^2-x+1) at x=0,
> I get the perfectly reasonable value f=1.
>
> But 1 is not in NaI, violating Moore's law.
Real analysis for { 1/(x^2-x+1) | x in [0,1] } does not lead to NaI. It's
optimal range enclosure is [1,4/3].
But in the example, you compute a non-optimal range enclosure by composition
of arithmetic operations. One of the intermediate operations is 1/[0,2].
This is violation of Moore's law since division by interval containing zero
is undefined.
> The zero in the interval is solely due to overestimation of the
> interval arithmetic.
I understand. It is to be expected because you compute the range enclosure
with a composition of arithmetic operations, and Moore's law applies to each
intermediate arithmetic operation as well.
In any case, dropping 0 from the domain of the intermediate step 1/[0,2]
gives [1/2,Inf), which is severely pessimistic.
Modal intervals improve the situation. Monotonicity gives:
fR(X) := 1/(X*(Dual(X)-1)+1)
over the monotonic domains x \in [0,.5] and x \in [.5,1]. So
fR([0,.5]) \union fR([.5,1]) = [1,4/3]
is the optimal range enclosure, and there is no NaI due to overestimation.
>>>>> I believe nothing in this motion and rationale hinders the
>>>>> implementation of various forms of non-standard intervals --
>>>>> Kahan, modal, etc. -- as discussed at the end of Vienna/1.2.
>>>> I've mentioned before this is simply not true. If traps or flags
>>>> are only way to obtain NaI result from an interval operation such
>>>> as 1/[-2,3], this is hinderance to efficient modal interval
>>>> implementations.
>>> This is another eason why modal intervals should not be part of
>>> the standard. It makes the latter unnecessarily complicated,
>>> only to introduce an error-prone technique that can be safely
>>> handled only by a tiny minority of users.
>>
>> I don't agree at all. It opens 1788 to a wider audience by
>> clarifying and simplifing.
>
> Modal arithmetic is a very dangerous tool that _easily_ leads to
> wrong results without a very good understanding of its theory.
I believe it is just a straw-man, Arnold. There have already been
discussions and examples in this forum of how it is a problem with intervals
in general. Classical endpoint analysis is particularly tedious and
error-prone, but the monotonicity theorems of modal theory simplify that
quite a bit.
>
> As our off-line discussion last year had revealed, not even you were
> able to interpret the modal theorems in the literature correctly to
> give always valid results.
Absolutely untrue.
It is unprofessional breach of confidence to drag elements of personal and
private discussions into a public forum like this. Especially to make such a
cheap and selacious point. I hope others that have spoken to you in
confidence will take warning by how you conduct yourself here.
Anyhow, I know what you are trying to distort and take out of context by
bringing this up, because I see the "trap" I believe you thought you could
corner me into by providing an example with x^2-x in it.
Between us, however, you know the example I give above for fR(X) shows your
claim about me is false, since according to your "understanding" of the
modal theory in our off-line discussions I should not have been able to
obtain an optimal result [1,4/3] for your new example.
>
> Thus the number of users that can safely handle modal arithmetic in
> full generality may well be zero.
IMHO, your failed attempt to "trap" me only emphasises how much you don't
know about the subject, and why your continued demagoguery against the
modal intervals is unjustified and irresponsible.
Nate