Re: Motion P1788/M0034.01:Notation-- discussion period begins
On 2012-04-23 08:43:50 -0500, Nate Hayes wrote:
> Hmm. I'd point out you and Vincent had basically argued the reason for
> unbounded intervals is they supposedly make developing computer algorithms
> (and proofs) at Level 1 much easier. But one doesn't need to look very hard
> to see this isn't the case. Interval Newton is arguably one of the most
> fundamental or important algorithms. At each step we have:
> X_2 := (m(X_1)-f(m(X_1))/F'(X_1)) \intersect X_1
> where m(X_1) is the midpoint of X_1. If the user provides an unbounded
> interval as input, the algorithm is undefined at Level 1 despite the fact it
> may operate well at Level 2.
Well, this example is a bit special as it also uses a numeric function
(f(m(X_1))). But if I understand the problem correctly (I'm not a
specialist of the interval Newton algorithm), I disagree with you:
the problem is not the unbounded intervals, but the fact that the
algorithm is not well stated. Indeed I suspect that you say it is
undefined at Level 1 only because the midpoint is undefined on
unbounded intervals. But if you replace "midpoint" by "any member of
the interval" (or perhaps something more restrictive), I think it is
well-defined at Level 1. Similarly, it is well-defined at Level 2 on
an unbounded input only if some arbitrary value is chosen for the
midpoint on such an interval.
If you meant something else, it would be better if you provided a
complete and detailed example.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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