Re: The current proposal
Nate Hayes schrieb:
I think the point Siegfried makes is that the "ideal" mathematical
properties of the interval arithmetic should drive the
implementation, not the other way around.
For example, the very simple formula for addition
[a,b] + [c,d] = [a+c,b+d]
breaks down if c=d=Inf and a=-Inf. In that case, IEEE arithmetic
produces
(-Inf,b] + (Inf,Inf) = [NaN,Inf),
which is indeed the current Intlab result. (No user of Intlab
ever had complained about this meaningless construct; so Siegfried
doesn't have to fear being grilled....)
I believe this is a reasonable result in a software implementation, as
it becomes very expensive to return something different.
It is quite cheap to convert (Inf,Inf) to NaN before doing the
operation. This settles the issue within the framework of the Vienna
Proposal.
And even though this proposal is not the standard to be, it
rebuts your claim that ''it becomes very expensive to return
something different''.
whereas the "ideal" result would be
(-Inf,b] + (Inf,Inf) = [0,Inf),
assuming the "infinity is number" paradigm where the infinity is not
a member of the interval but rather a token for an unbounded real
endpoint.
So (Inf,Inf) is an interval with two unbounded real endpoints???
I have no idea what this could possibly mean.
There is nothing that makes the result [0,Inf) plausible.
(The apparent assumption that the Inf's have the same huge finite real
value is completely unjustified.)
Assuming the infinities are the same unbounded real value is justified.
It follows naturally from correctly rounded floating-point arithmetic.
It does not follow naturally.
1/[-1,0] + 1e400 should give in exact arithmetic [-inf,1e400],
hence with infinity-as-number [-inf,inf] in verified floating
point arithmetic.
If you don't allow the operation 1/[-1,0] to give the value
[-inf,0] then the division operation is not appropriate for
applications to global oto=imization.
But if you assume 1/[-1,0] = [-inf,0], your proposal gives the
severe underestimate [0,inf].
Even the Vienna Proposal says so.
???
In the Vienna Proposal, Inf is never a real number, and [Inf,Inf]
is always converted into the empty interval. This is even the case
in Section 7, where compatibility issues with IEEE directed rounding
is discussed.
Arnold Neumaier