Re: The definition of cancel_minus
Dear Evgenija
I read your paper, that was very good.
In the near future I would like to put an interesting problem on the
table, as well as the lost silver bullet, which may be interesting,
especially for J. Peryce.
Warmest regards,
On 9/20/16, Evgenija Popova <epopova@xxxxxxxxxx> wrote:
> Walter and all,
>
>> What is the rational behind the standard's choice?
>
> I suppose that the reason has been: all operations defined by the
> Standard are outwardly rounded.
>
>> In which (practical) use cases this difference
>> in Definitions I and II would be relevant?
>
> Both outwardly rounded and inwardly rounded operations are used in the
> applications.
>
> In general, these operations have the property that inwardly rounded
> result can be obtained by outwardly rounded operations (and vise versa)
> in a particular way (see doi:10.1016/j.cam.2005.08.048 formulae (8), (9)).
> The problem is that the cancelPlus/Minus operations defined by the
> Standard are half of the actual “inner” operations defined in Motion 12
> “Inner Addition and Subtraction” (see
> http://grouper.ieee.org/groups/1788/private/Motions/Motion12.02.pdf)
> which was the background of these operations. This hampers the
> applications (as discussed in a recent paper DOI:
> 10.1007/s00419-016-1180-2).
>
> I use the present discussion to announce the latter recent paper DOI:
> 10.1007/s00419-016-1180-2 (free access at http://rdcu.be/kofz) which was
> motivated and clarifies the discussion raised by Mazandarani,M.: IEEE
> standard 1788–2015 vs.multidimensional RDM interval arithmetic, in this
> working group in Sept./Oct. 2015, see
> http://grouper.ieee.org/groups/1788/email/msg08439.html.
>
> Best regards,
> Evgenija Popova
>
>>
>> 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.
>>
>>
>
--
Mehran Mazandarani
Department of Electrical Engineering
Ferdowsi University of Mashhad, Mashhad, Iran.
homepage:http://mehran.mazandarani.fumblog.um.ac.ir/
IEEE Member, me.mazandarani@xxxxxxxx