On 2012-04-26 10:03:40 -0500, Nate Hayes wrote:
Vincent Lefevre wrote:
On 2012-04-25 09:05:19 -0500, Nate Hayes wrote:
A few points:
-- No computer (that I'm aware of) can numerically prove hardly
anything
useful about the domain of a function beyond the underlying numeric
limits
of the system; so for this reason alone, truly unbounded intervals are
never
necessary in numeric models or computations (you have never answered
my
original question from long ago to show a counter-example of this).
I disagree. If the user asks for the range of 1/[0,1], the math result
is [1,+inf]. This is useful information, at least much more than an
error.
You've avoided the question again.
I'm not avoiding it. This is a perfect answer to your question.
Others have given other examples.
The question is, what is an example of an interval algorithm, for
example, that proves all the zeros of a function on the domain
[MAXREAL,+Inf].
This question is pointless. If there is an unbounded interval in
the problem, then you need unbounded intervals to be able to express
the problem!
Algorithms will obviously depend on the function, on the implementation
parameters (e.g. whether you have enough precision or not), and so on.
There are well-known techniques such as scaling (look at the MPFR code,
for instance). There are also implementation limits.
Vincent Lefevre wrote (cont.):
You may build a theory where the computed result as a *family* of
intervals, but then you should stop calling that an interval. This
is obviously more complex (as a specification) than unbounded
intervals and I don't see what it would bring.
During the course of this discussion, it appears your position changed
from
unbounded intervals are "essential" to "they make algorithms more easy
to
define at Level 1." But my observations are that both these statements
are
not true.
It has not changed. Simplicity is essential.
for this reason it is possible to define
midpoint([1,+OVR]) at Level 1 in the same way P1788 is currently
considering to do so at Level 2.
How would you define it at Level 1, as a *real number*?
At Level 2 P1788 has considered to define midpoint([1,+Inf]) =
(1+REALMAX)/2
or something similar.
No, it is REALMAX, if one really wants the midpoint to be defined.
At Level 1, midpoint([1,OVR]) could for the same reasons be
similarly be defined (1+H_f)/2.
This would be bad, even if you define it as H_f (a bit like in Level 2),
because at Level 1, you have intervals like [1,4*H_f], whose Level 1
midpoint is 2*H_f+1/2> H_f. Thus conventional inequalities (see the
discussion about midpoint) would no longer hold at Level 1.
-- Replacing "midpoint" with "any member of the interval" gives a
valid
mathematical definition of the Interval Newton, but such a definition
is
also then no longer an algorithm because the exact method of choosing
"any
member of the interval" is left undefined.
Well, once the one who writes the algorithm choose the member in
question, he has an algorithm. This member can potentially be a
parameter, and it will still be an algorithm. I don't see any problem
with that!
The problem is one cannot choose midpoint as this parameter at Level 1
without causing the Interval Newton to be undefined when unbounded
interval
is provided as input.
This is not a problem. You don't necessarily need a midpoint at
Level 1 for this algorithm.
You are the one who said unbounded intervals are important to ensure
algorithms are easy to define at Level 1. But this is clearly not
always the case.
The problem is just with your reasoning.