Re: Structured support for both standard and modal intervals
Nate
On 24 May 2012, at 18:10, Nate Hayes wrote:
> I only have one further comment (see below).
>
> John Pryce write:
>> Nate Hayes wrote:
>>> P1788 already voted that the midpoint of an unbounded interval is
>>> undefined.
>>> So if P1788 decides there will be no overflow, then these are my
>>> lingering
>>> and unanswered questions:
>>>
>>> 1. If arithmetic mean (midpoint) of an unbounded interval is undefined
>>> at Level 1, does this mean other interval bisection methods such as
>>> geometric mean, smedian2, etc. are also undefined at Level 1 for
>>> unbounded
>>> intervals?
>>
>> IMO it's right that "midpoint" should be defined entirely "by the
>> mathematics", so undefined...
>>
>> My view of undefinedness. The Hack-Neumaier flmedian is inherently
>> undefined at Level 1. The Pryce-Zuras smedian at Level 1 is undefined, or
>> infinite, for unbounded intervals. Who cares? They may be found
>> pragmatically useful at Level 2.
>>
> ...
>>> 3. If answer to (1) is "yes", then are all of these interval bisection
>>> methods undefined for unbounded intervals at Level 2, as well?
>>
>> For me this a matter of mathematical honesty. If ANY numeric function is
>> mathematically undefined for some input then say so, at Level 1; and let
>> it return NaN at Level 2...
>>
>>> 5. If answer to (3) is "yes", it seems users will not be able to write
>>> any interval bisection methods of thier own that will be conforming,
>>> since
>>> picking arbitrary real number will be against the Level 1 and Level 2
>>> definitions. So how can any user develop conforming algorithms that
>>> bisect
>>> unbounded intervals?
>>
>> If you CALL IT SOMETHING DIFFERENT, you can make your pragmatic function
>> do whatever is useful. So no contradiction...
>> I should probably add to the Level 1 text, that other operations may be
>> defined at lower levels for purely pragmatic reasons.
>
> My only problem with this idea is it means users can no longer design
> interval algorithms (or prove them correct) at Level 1, and we feel it encourages users to ignore the Level 1 definitions. We believe overflow avoids this problem by providing the necessary mathematical framework at Level 1a.
My take on this is that it is not so, because the magic word "some" (arbitrary) can be and should be used in the definition of many algorithms at Level 1:
- In validated solution of ODE initial value problems x'=f(t,x), if I remember right, one has a box X enclosing a set of solutions at time t. One chooses "some" point xhat in X and expands the (point) Taylor series about that point, with a remainder term to account for the size of X and for rounding errors. It is usually best to take xhat as the midpoint of X, but the algorithm is correct whatever xhat is chosen.
- In your B&B method, one can choose "some" point (in "some" side) of a box to split it at. One can prove the method converges in the sense that each side of each box eventually becomes smaller than a tolerance TOL, provided certain constraints are satisfied. First, I would map the real line to a bounded interval by a map such as x -> y = x/(1+|x|) so as to get a meaning of "small" that works at infinity as well as at finite points. Then the constraint might be that, if (in this y coordinate system) a side is split in the ratio c:d where 0 <= c <= d, c+d=1, then c must >= some fixed c0 > 0.
- Similarly, isn't interval Newton valid for an arbitrary "expansion point" in the current interval?
A correctness proof that is independent of any specific heuristic for the "some" is often useful.
John Pryce