Re: Motion P1788.1/M004.01
P1788.1 members
> On 10 May 2016, at 17:17, Lee Winter <lee.j.i.winter@xxxxxxxxx> wrote:
> On Tue, May 10, 2016 at 10:46 AM, Ulrich Kulisch <ulrich.kulisch@xxxxxxx> wrote:
>
>> I think interval arithmetic should not be defined over the IEEE 754 binary64
>> numbers. This more or less pulls all the IEEE 754 exceptions into interval
>> arithmetic. We shoud not bother all users of interval arithmetic with
>> constructs which really do not occur and definitly are not needed in
>> interval arithmetic.
>>
>> Interval arithmetic shoud just be defined as a calculus for connected sets
>> of real numbers.
>
> I disagree. We don't have native real numbers. We only have finite
> (small) sets of rational numbers.
>
>> Since -oo and +oo are not real numbers they cannot be
>> elements of a real interval.
>
> Agreed. However, while we CALL them signed infinities, they are not.
> In point of fact they are actually signed values for a result that
> somewhere in the preceding calculations exceeded the maximum
> representable value in the applied FP encoding .
The 1788 standard *is* based on the mathematical real numbers, as the text of Level 1 makes clear. I agree with Ulrich in that this is how it *should* be, and only disagree with him in that he thinks it isn't — but it is. Oh yes, and that there are genuinely infinite ±oo.
If one wants guaranteed answers to *mathematically stated* questions like "Does this PDE with these boundary conditions have a negative eigenvalue?", or "how many homoclinic orbits does this dynamical system have?", this is the sort of foundation one has to use.
If I understand Lee Winter right, he is more interested in analysing the behaviour of actual finite-precision programs. I have much respect for his view of intervals, but it is based on a different mathematical foundation from 1788.
This is not compatible with the 1788 set-based flavor. Could it be a 1788 flavor of its own? That would be interesting.
John Pryce