Dear colleagues,
I am implementing the function cancel_minus
and I do not understand the rational behind
its definition in the standard. In the "normal case"
(ie. non empty and bounded intervals) section
10.5.6 of the standard states that
Definition I: cancel_minus(sum, parcel) = tightest interval c
such that parcel + c contains sum.
In my opinion, the definition should be
Definition II: cancel_minus(sum, parcel) =
(the convex hull of) the union of all intervals c
such that parcel + c is contained in sum.
Definition (II) is what we need if we want
to recover the interval c when are only given
a computed sum and a parcel, as in
the example mentioned in the standard:
we are given a computed sum = SUM_{k=1}^n a_k
and a_j and want to recover SUM_{k != j} a_k.
In practice, given only the parcel and the sum,
we only know that a candidate c to
cancel_minus(sum, parcel) satisfies
"parcel + c is contained in sum",
because, in practice, sum will be slightly
larger than the exact sum, and this is what
I would want cancel_minus to model in theory.
Therefore, it would be natural to take
cancel_minus(sum, parcel) as the union of all
such c (and take the convex hull just in case
this beast is not convex,) Both in theory
and in practice the actual c used
to generate the sum is certainly contained in
cancel_minus(sum,parcel) in Definition II.
Definition II has the advantage that, at level 1,
the resulting cancel_plus is always well defined,
because the empty interval e satisfies
"parcel + e is contained in sum" and the
union of intervals in this definition is taken over
a non empty family of intervals.
Definition II differs significantly from Definition I
when sum is shorter than parcel, and they are both
not empty. In this case, at level 2, Definition II
leads to an empty cancel_minus whereas the item
12.12.5 in the standard yields cancel_minus = entire.
What is the rational behind the standard's choice?
In which (practical) use cases this difference
in Definitions I and II would be relevant?
Following Definition II we could derive
the results in non normal cases. For
instance, we would obtain, at level 1,
that cancel_minus(entire, parcel) = entire. This
result is natural but in conflict with what is
described in item 10.5.6 of the standard,
but this time the results match at level 2.
With Definition II we can also handle
meaningfully half infinite intervals, as in
cancel_minus([0,+oo], [0,1]) = [0,+oo],
whereas the standard leads to
the less desirable result
cancel_minus([0,+oo], [0,1]) = entire
at level 2.
What points am I missing?
regards,
walter.