Two kinds of interval arithmetic domains
Reading these last few discussions on whether an IA standard should
address experts or naive users (presumably both, though perhaps in
different ways), it occurred to me that there are at least two rather
different ways that IA might be used, and that the intuitions in the
two cases may be very different.
The first domain is that of reliably-bounded computations with uncertain
numbers. In this domain all intervals are expected to be bounded and
relatively narrow. If intermediate interval results become unbounded,
or perhaps even exceed a settable uncertainty threshold, it might be
useful to stop the computation and look out for a different approach:
use a different algorithm, retry with higher precision, or give up and
report an ill-conditioned problem. This is the domain where MidRad
representations may have many advantages, and where the precision of
the bounds loses its significance (pun intended) as intervals become
too wide to be useful.
The second domain is that of containing results that depend on parameters
within certain ranges. Any given parameter value may be highly precise,
but is picked from a bounded (or semi-bounded) range. This is the domain
or forwards and backwards containment evaluations, and of the various
flavours of "non-standard" intervals. Here the precision of the bounds
can be critical, e.g. in the vicinity of singularities -- sometimes it
even matters whether the bounds themselves (as FP numbers) are considered
to be included or excluded.
The two have much in common of course -- in particular, the techniques of
the second domain can be used (by experts) to derive algorithms to help
users in the first domain (which ought to be accessible to beginners).
Any thoughts on this aspect of things?
Michel.
---Sent: 2009-04-19 16:11:55 UTC