Re: The current proposal
Nate Hayes (via Baker Kearfott) wrote:
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....)
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.
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.)
For a naive infinity-as-number interpretation, the result should be
the hull of all sums of elements in the arguments, hence should be the
hull of NaN and inf. Nobody has so far suggested a well-motivated
definition of what this hull should be.
Under Cset arithmetic, the result would be [-Inf,Inf].
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Nate Hayes wrote (in another mail):
The point is that always, without case distinction, the following
should be true:
(1) For a given function F, for example, X=F(interval(A.sup)) should
give an
inclusion of the value of F at the right bound of A.
Always???
This value is not even properly defined in many cases,
such as when F(x)=x-x or F(x)=(x-1)/(x+1) and A=[0,inf].
If "infinity as number" is true, i.e., if the infinity is not a member
of the interval but rather a token for an unbounded real number,
What is an unbounded real number?
I have never heard of such an object being given a mathematically
valid definition.
then it
is properly defined:
F(Inf)=Inf-Inf=0
F(Inf)=(Inf-1)/(Inf+1)=Inf/Inf=1
Arnold Neumaier