Re: MidRad to/from InfSup (was: the "set paradigm" is harmful)
Shin'ichi Oishi schrieb:
what I proposed is
set round up
m=0.5*(l+u)
r=m-l
This gives a correct enclosure but has the blemish that
m need not be in [l,u].
E.g., in 3-digit decimal arithmetic when l=5.20, u=5.21,
we have l+u=10.5, m=5.25, r=0.05.
In my opinion, m in [l,u] is important to avoid unintended
surprises since the midpoint might be later used as a typical
point of the original interval.
----- Original Message ----- From: "Michel Hack" <hack@xxxxxxxxxxxxxx>
Hovever, 1.001 +- 0.00011 does not cover the original lower bound,
as the exact range is 1.00001 to 1.00111 -- it is slightly shifted.
I conclude that the proposed formula does not always provide
a valid MidRad enclosure of a given InfSup interval.
Now I'll open my proposed alternative to attack; my head is spinning:
Set round nearest
m = 0.5 * (l+u)
Set round up
r = max(u-m, m-l);
Sorry for this sloppiness in the Vienna Proposal.
I think the formula that should be in the standard is
set round up
m=l+0.5*(u-l);r=m-l
This ensures that m in [l,u] for any basis, and gives in the above
example the enclosure m=5.21, r=0.01, which is 5 times narrower than
the Oishi enclosure.
In binary arithmetic, or when intermediate results are kept to higher
precision, the two formulas appear to be equivalent.
Arnold Neumaier