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In addition, as I mentioned in my posting of 09 Feb 2009,
both domains should be extended to complete
(Kaucher/modal) interval arithmetic, which can be invisible
for naive users.
I agree with Markov on this point. For people that are interested in the
mid-rad arithmetic, providing a Kaucher/modal representation of the mid-rad
intervals (see, e.g., Markov's SCAN 2006 paper) is very important.
Mid-rad calculations are inherently pessimistic to begin with. They are also
bound by the constraint that Rad <= 0.4 |Mid|. So for these reasons they can
be very sensitive to interval dependence.
The theorems of Gardenes, at. al., may allow Kaucher/modal mid-rad intervals
to compute optimal and/or narrowed range enclosures for such calculations.
In many cases, this may prevent interval dependence from causing the Rad <=
0.4 |Mid| constraint to be violated, therefore allowing a computation to be
done entirely in mid-rad arithmetic that might otherwise need to be finished
in inf-sup form.
Of course, practical implementation of the Kaucher/modal mid-rad arithmetic
comes basically for free over the classical mid-rad arithmetic, just as it
does for the inf-sup form of intervals.
Nate Hayes
Sunfish Studio, LLC
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