Re: MidRad to/from InfSup (was: the "set paradigm" is harmful)
Svetoslav and P1788:
Yes, IF we decide to include midpoint-radius (something that
should be formally decided before we formally decide on HOW
to include them), we may want to avoid specifications of
edge cases. (I haven't seen a formal second to the motion
to include midpoint /radius, unless I have missed something.)
If we avoid "edge cases," specifying what is a "narrow" interval
might be as messy as including edge cases. A position paper indicating
more precisely how this would be done might resolve the issue. We
could then put that position paper forward for a vote, if we get
a second for the motion to include midpoint-radius, and if that
motion then passes as a position according to our 60% rule.
Baker
Svetoslav Markov wrote:
IMO one should not worry about specifying midrad fp-operations
for edge cases involving wide intervals. This is simply useless.
Midrad interval arithmetic is intended for narrow intervals in order
to model computations with approximate numbers.
One only has to define what is a narrow interval. Generally speaking those
are intervals for which no unpleasant effects (of the sort that Arnold
mentions) exist.
I think that the first such unpleasant effect appears when r>(sqrt 2 - 1) m,
since then the centred square of a positive interval contains negative
numbers. In practice this means that an interval is narrow, if r<= 0.4 m.
Restricting computations to narrow intervals will mean in practice that results
should be checked at each step for narrowness. However, this can be an
extremely fast process as it can involve only the exponents of r and m. This,
combined a 1-2 digit mantissa for r will lead to a very fast midrad arithmetic.
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R. Baker Kearfott, rbk@xxxxxxxxxxxxx (337) 482-5346 (fax)
(337) 482-5270 (work) (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
Box 4-1010, Lafayette, LA 70504-1010, USA
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