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Re: Unbounded intervals

On 04/26/2012 05:30 PM, Vincent Lefevre wrote:

On 2012-04-26 10:03:40 -0500, Nate Hayes wrote:



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.
The problem is just like the problem of finding all zeros of a function in [1,1+eps] if the function is highly oscillatory there.
One either needs more resolution to resolve the problem (which can be done with multiprecision arithmetic), or one accepts the interval as the solution (which is the common way to do it).
[MAXREAL,+Inf] is no difference in this respect; the interval is only one ulp wide (though the ''last place'' in this definition is somewhat singular), and returning such an interval to the user is precisely what a tyoical user would expect to see if there are solutions close to infinity. It amounts to an excellent enclosure in the projective completion of the real line. 


Arnold Neumaier