On 3/8/07, Jay Belanger <belanger at truman.edu> wrote:
> "Stavros Macrakis" <macrakis at alum.mit.edu> writes:
> > Incidentally 1^inf => und differs from IEEE 754 since 1^inf => 1
> there.
> >
> > I would be interested to understand why IEEE 754 did this. It looks
> like a bug to me,
> > since (1-epsilon)^inf = 0 and (1+epsilon)^inf = inf, for arbitrarily
> small positive
> > epsilon.
>
> I'm don't know why IEEE 754 did this, but it looks reasonable to me,
> since 1 is not 1-epsilon or 1+epsilon. I think it would be reasonable
> to treat 1^inf and 1.0^inf differently, and let 1^inf=1 and
> 1.0^inf=nan or und.
Well, IEEE 754 is about floating-point, so 1^inf above was shorthand for
1.0^inf.
Now here's the problem in Maxima. Unless we do a global analysis of every
expression (what Limit does), then expressions will simplify in the usual
bottom-up way. What's more, the expression may only be built piece by
piece, so the whole expression may no longer be accessible.
So if you start with an expression like (1-1/inf)^inf, it will simplify to
(1-0)^inf then to 1^inf, which it is being proposed should simplify to 1.
But following the semantics I believe we all agree on for expressions with
multiple inf's, the original expression must be interpreted as limit(
(1-1/x)^y, [x,y], inf), which is IND. Proof: consider various x's and y's
which go to inf:
(1 - 1/exp(q))^q => 1
(1- 1/q) ^ exp(q) => 0
So simplifying 1^inf => 1 gives the wrong result.
Now I suppose we could try to keep track of infinitesimals, so that 1/inf
simplifies to zeroa, and 1-zeroa doesn't simplify at all, etc. That would
be an acceptable, though more complicated solution. Until and unless we do
that, I think we are forced to simplify 1^inf to UND.
Stepping back a bit and being a bit more philosophical, what seems to be
happening here is that as soon as you start manipulating infinities as
numbers, you had better think about infinitesimals as well if you don't want
to fall into contradictions. An approximation to a full infinitesimal
analysis treats every finite number as being "surrounded" by its
infinitesimal neighborhood.
-s