> Date: 20 Nov 90 00:30:55 GMT
> From: ····@data.UUCP (Ken Dickey)
> Has anyone out there done an implementation of reals using interval
> arithmetic?
Many years have gone by an nobody ever responed to you, until now.
(I discovered ykur article just today when doing a Google search.)
The real numbers are a field consisting of equivalance classes of
Cauchy sequences, or Dedekind cuts, etc. What aspects of these
definining characteristics of real numbers would you like emulated? If
you emulate real numbers by infinite sequences of intervals where each
interval contains the next as a subset and there is no lower bound on
the length of the intervals within any such sequence (i.e. the limit on
interval length approaches zero as the term index approches goes
further and further out toward +inf), then variable-precision interval
arithmetic sorta emulates real numbers, the "sorta" being that only
finite precision is available on any given computer system.
··········@YahooGroups.Com wrote:
>> Date: 20 Nov 90 00:30:55 GMT
>> From: ····@data.UUCP (Ken Dickey)
>> Has anyone out there done an implementation of reals using interval
>> arithmetic?
>
> Many years have gone by an nobody ever responed to you, until now.
> (I discovered ykur article just today when doing a Google search.)
> The real numbers are a field consisting of equivalance classes of
> Cauchy sequences, or Dedekind cuts, etc. What aspects of these
> definining characteristics of real numbers would you like emulated? If
> you emulate real numbers by infinite sequences of intervals where each
> interval contains the next as a subset and there is no lower bound on
> the length of the intervals within any such sequence (i.e. the limit on
> interval length approaches zero as the term index approches goes
> further and further out toward +inf), then variable-precision interval
> arithmetic sorta emulates real numbers, the "sorta" being that only
> finite precision is available on any given computer system.
As it turns out, I've long since solved the problem by myself. But thanks
for the offer!
--
No, I'm not really who I'm pretending to be. :P
I have been experimenting with writing an interval library for Common Lisp
(FYI, I am a lisp newbie). There are some challenges but I think lisp
provides the ideal environment for exploring interval computations.
Something to note about computations with intervals. It is
generally difficult to get optimal bounds on an expression,
for example, X * (Y + Z) does not necessarily equal X * Y + X * Z,
where X,Y,Z are intervals on the real line. A demonstration of this is
given below using my home brewed interval library:
CL-USER>(load "limits") ; my interval library hack
(setf X #[-2 1])
#[-2.0d0 1.0d0]
CL-USER> (setf Y #[-3 -1])
#[-3.0d0 -1.0d0]
CL-USER> (setf Z #[2 5])
#[2.0d0 5.0d0]
CL-USER> (i* X (i+ Y Z))
#[-8.0d0 4.0d0]
CL-USER> (i+ (i* X Y) (i* X Z))
#[-13.0d0 11.0d0]
Interval computations are always conservative. There are a range of
techniques to handle this sub-optimality, but in general estimating the
exact range for an arbitrary rational interval function is an NP-Hard
problem.
If any one is interested in interval arithmetic you can start from my page
http://creativelimits.net/research/interval/ or from Vladik Kreinovich's
excellent site http://www.cs.utep.edu/interval-comp/
I am excited to see an interest in interval computations on c.l.l and c.l.s
news groups.
-Janos
··········@YahooGroups.Com wrote in message news:<·················@Yahoo.Com>...
> > Date: 20 Nov 90 00:30:55 GMT
> > From: ····@data.UUCP (Ken Dickey)
> > Has anyone out there done an implementation of reals using interval
> > arithmetic?
>
> Many years have gone by an nobody ever responed to you, until now.
> (I discovered ykur article just today when doing a Google search.)