Re: The definition of cancel_minus
prati mi statiite, molia, s
On 20 Sep 2016 at 18:38, Evgenija Popova wrote:
Date sent: Tue, 20 Sep 2016 18:38:55 +0300
From: "Evgenija Popova" <epopova@xxxxxxxxxx>
To: Walter Mascarenhas <walter.mascarenhas@xxxxxxxxx>,
Stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
Send reply to: epopova@xxxxxxxxxx
Subject: Re: The definition of cancel_minus
> 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.
> >
> >