Re: The current proposal
Arnold Neumaier schrieb:
Nate Hayes schrieb:
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.
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].
No.
In your example, zero is a member of the denominator, so infinity is a member of the result produced by the reciprocal operation. If zero is not a member of the denominator, then
1/[-1,0) + 1e400 = (-Inf,-1] + (Inf,Inf) = [0,Inf).
If intervals [a,b] and [c,d] are entirely real, i.e., if they do _not_ contain infinity, then the formula
[a,b]+[c,d]=[a+c,b+d]
specifies all that topologically matters for the range enclosure. In this case, a and c are real numbers with the same very large magnitude (they represent overflows of the digital scale), so the formula is correct.
C-sets must return [-Inf,Inf] because they allow the infinity to be member of the intervals, and this destroys some of the algebraic properties of the real numbers. Hence the formula [a+c,b+d] is not enough for proper range enclosure. This
is why c-sets must return the wider result.
Nate Hayes
Sunfish Studio, LLC