the "set paradigm" is harmful
Svetoslav Markos wrote:
> It has been shown that intervals presented in midpoint-radius
> notation satisfy the same algebraic structures as intervals
> using inf-sup presentation. The argument of some supporters
> of the set-paradigm that an interval of the form [a, oo)
> cannot be presented in midpoint-radius form seems to me
> ridiculous.
Really?
The ability to represent unbounded intervals permits InfSup
arithmetic to be exception-free: overflow simply results
in an unbounded interval -- and not necssarily Entire.
I suppose MidRad could do something similar -- but it would
have to abandon BOTH bounds and return Entire (Infinite
Radius, undetermined Midpoint). Is that how it's done? I
would have thought that an exception would be preferable.
It seems strange that the square of a big approximate number
could be an enclosure that includes all negative numbers (as
well as all positive numbers).
Let's consider representable entities more carefully. Assuming
for now the same precision (and radix) for all four components
(Inf, Sup, Mid, Rad), what is the blowup when converting from
one form to the other? Let's restrict this to bounded intervals.
MidRad can represent some very narrow intervals around certain
numbers (namely floating-point numbers), whereas non-point InfSup
intervals have a width of at least 1 ulp. This could be useful,
as this set includes small integers and simple dyadic rationals
(even in decimal). When converting such a narrow MidRad interval
to InfSup, the blowup could be very large.
MidRad can also represent intervals that exceed the range of
InfSup bounded intervals, as Mid+Rad can exceed MAXREAL. Again,
converting such a MidRad to InfSup would result in an unbounded
interval, hence with infinite blowup.
Except for these extreme cases, MidRad can be converted to InfSup
with minimal blowup of at most two ulp. In base 10 this could be
relatively large however, when the interval contains a power of the
base. (The relative error can increase by a factor of the base.)
For bounded intervals, conversion from InfSup to MidRad is typically
less lossy (i.e. less blowup) -- typically a bit more than half an
ulp, but not more than one ulp. However, a strictly positive interval
could end up having a zero endpoint: when Inf is much smaller than Sup,
and both positive, we would get Mid = Rad = ceil((Inf+Sup)/2). This
could change the dynamics in a significant way down the road.
I think back-and-forth conversion would be stable after one round.
So -- how ARE overflow and unbounded intervals handled in MidRad
arithmetic?
Michel.
Sent: 2009-02-10 23:11:52 UTC