On Fri, 25 Jan 2008 05:32:44 -0800
"William Stein" <[EMAIL PROTECTED]> wrote:
> On Jan 25, 2008 3:36 AM, Robert Bradshaw
> <[EMAIL PROTECTED]> wrote:
> >
> > I agree--rather than just having a signed/unsigned infinity ring, I
> > think there should be an extended ring for Z, Q, R, C--both signed
> > and unsigned when there is an absolute order. When does the concept
> > of infinity make sense for a given ring? 1 (or 2)-point
> > compactification? (not for Q), ordering (not for C), maybe an
> > unbounded valuation (making "p-adic infinity" equal to zero?)
> >
> > Implementation-wise, this could be accomplished via a generic
> > wrapper around the rings and their elements (which I think could
> > still be quite fast, and coercion would be natural and efficient
> > too, not to mention the difficulties and inefficiencies raised by
> > having multiple types for a single parent, or Integers with
> > non-integer-ring parents).
> >
> > The default sage oo would be +Infinity in Z \union {\pm \infty}.
> >
>
> Yes, this sounds good.
>
> I -- then David Roe -- didn't implement something like the above I
> think only because we didn't need it for our applications (to modular
> symbols and p-adics). But I could certainly see how it would
> be very useful for more general applications.
I think the current approach is good. It handles the corner cases where
infinity comes up efficiently. There is no need to keep track of
multiple types from the same parent, check every time arithmetic is
done, or to further complicate coercion.
The problem could be solved easily by returning 0 (a numeric zero) when
a finite value is divided by infinity. Then, the coercion model would
take over, and things would work again.
i.e. with the proof of concept patch below, the following works
sage: t = UnsignedInfinityRing.gen()
sage: 1/t
0
sage: 1 + 0/t
1
sage: 1 + 1/t
1
sage: 1 + t
Infinity
Instead of implementing extended integers, rationals, reals or complex,
and complicating, as well as slowing down basic arithmetic, could we
just allow these infinity rings to behave not so algebraic? (and maybe
change their name and not call them a ring?)
Burcin
Here is the patch:
--- a/sage/rings/infinity.py Fri Jan 18 15:25:23 2008 -0800
+++ b/sage/rings/infinity.py Fri Jan 25 23:16:42 2008 +0100
@@ -263,7 +263,7 @@ class LessThanInfinity(RingElement):
def _div_(self, other):
if isinstance(other, UnsignedInfinity):
- return self
+ return sage.rings.integer.Integer(0)
raise TypeError, "quotient of oo by number < oo not defined"
def __cmp__(self, other):
--~--~---------~--~----~------------~-------~--~----~
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