Siegfried's recent paper, and other matters
Siegfried and all
On 31 Dec 2011, at 07:05, Siegfried M. Rump wrote:
> On Fri, 30 Dec 2011 14:56:26 -0100, John Pryce <prycejd1@xxxxxxxxxxxxx> wrote:
>> I think Siegfried Rump's recent paper is a great piece of work, whose implications I am still absorbing. It could well lead to a practical interval arithmetic that handles overflow and underflow better than we currently can. But ... its time has not yet come.
>
> The arithmetic is based on a mathematical theory given in the paper. Properties are proved which IMHO are not valid for other definitions of interval arithmetic. The difference to what everybody knows is the handling of huge and tiny numbers. I think it has been worked out in detail, and it is easy to grasp.
Well, my - and Bill Walster's - cset interval arithmetic also is based on solid, published mathematical theory, and I made a sample implementation of it (based on Intlab). It overcomes various defects of traditional IA, and I believed at one time it was the IA wave of the future.
But Arnold, Nate and others say, on the basis of their practical experiments, that its virtues are balanced or outweighed by various vices.
So, certainly for this standard, we should be cautious of new theory. In Hilaire Belloc's words,
> And always keep a-hold of Nurse
> For fear of finding something worse.
However, as time permits I intend to grasp the Siegfried theory in depth. Maybe it has a cset version...?
As for modal (Kaucher) intervals, my view is:
1. They should not be part of the main standard, but see item 4. A necessary consequence of this is a "two library solution", one for ordinary ("proper" in modal terminology) intervals and one for modal intervals - say ORDINARYINTLIB and MODALINTLIB.
2. The demands of intervals-as-sets shall always take precedence over other interpretations.
3. However, we should exercise due diligence to eliminate, as far as practical, elements of the standard (whether required or recommended) that prevent modal interval operations being a straight extension of those for ordinary intervals. According to recent remarks by Nate, much of this can be achieved by choice of names, e.g.
- ordinary "intersect" to be a separate operation from modal "meet";
- separate constructors for ordinary and modal intervals.
A desirable aim, IMO, is
A program that runs correctly with ORDINARYINTLIB shall
also run correctly, with NO changes, with MODALINTLIB.
4. A modal interval standard should be included as an Informative appendix to the main standard. I will not write the text of it. That is the responsibility of the modal folk, who already have a subgroup for the purpose. However, when the text comes I will do my editor's job of integrating it into the rest of the document.
I believe items 1 to 3 of the above represent essentially the view Arnold Neumaier took in the Vienna proposal, and that item 4 is implicit, and maybe explicit, in earlier discussions.
Regards
John Pryce