Re: the "set paradigm" is harmful
Yes, as Baker suggested, I would like to make the motion as the follow:
An interval can be represented in two ways in the IEEE-1788 standard. One of them is its lower and upper bounds as [inf, sup]. The other is its midpoint and radius as {M, R}. When the rounding error is ignored, the transformation between these two representations are: M = (inf + sup)/2, and R = (sup - inf)/2; and inf = M - R. and sup = M + R. To ensure enclosure property in the [inf, sup] form, transformation from {M, R} form to [inf, sup] form must consider rounding mode properly in implementation.
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The rationales for this motion have been discussed lately and in Baker's comments.
Chenyi Hu
>>> Ralph Baker Kearfott <rbk@xxxxxxxxxxxxx> 2/10/2009 7:43 AM >>>
Svetoslav et al,
I see several possibilities of accommodating midpoint-radius
representations in the standard. One might be to simply
specify two representations for intervals, and specify
conversion between the two. (That might be the simplest way.)
Does anyone care to make a formal motion to that effect?
If such a motion passes, we can then work out details.
There is another representation of intervals in use to avoid
having to change the rounding mode: [-inf,sup]. However, we
may be able to accommodate that representation within the [inf,sup]
representation by "as if" wording.
Baker
P.S. I'm not sure in exactly what places, if any, the issue of accommodating
midpoint-radius impacts the two motions currently being
formally processed. (P1788/M0001.01_StandardizedNotation and
P1788/M0002.01_ProcessStructure) The first deals with the
notation to be used in the standards document and the second deals
with the structure of the standards document (and not its actual
content). If there is anything within these motions that
impacts midpoint-radius, please be VERY specific about where.
On 2/10/2009 3:53 AM, Svetoslav Markov wrote:
>
>
>
> On 9 Feb 2009 at 9:19, Michel Hack wrote:
>
>> Earlier I wrote:
>> ... when the radius exceeds either an absolute or a relative threshold.
>>
>> Change "either" to "both". One applies for small absolute values, the
>> other for the rest.
>>
>> Btw, I realise that one can also have an approximate number arithmetic
>> with containment properties, and that this is probably what Svetoslav
>> had in mind -- but my comments about restriction to narrow intervals
>> remain applicable.
>>
>> Michel.
>
>
> Surely, MR-presented intervals, resp. approximate numbers,
> admit probabilistic interpretation, e.g. midpoint can
> be mean value, in Gaussian distributions radius can be
> standard deviation or half-width of confidence interval, etc.
> (Note that the guaranteed containment property can be still there,
> say within the 95% of probability). Such interpretations depend on the
> particular models/problems. MR-presentation supports various
> interpretations as summarized in Arnolds classification. Note that
> MR-form intervals incorporate the set paradigm --- up to the
> special cases (oo, a], [b, oo). That is why MR-form is even more
> important than sup-inf form. MR-form is more convenient than the
> sup-inf form in many applications (possibly for Taylor polynomials
> as well). Some authors often use this form as an intrinsic form in
> computations/proofs and then translate results in sup-inf form
> (notably J. Rohn). This form incorporates and enhances several
> suggestions made by U. Kulisch.
>
> Conclusion is: MR-presentation of intervals deserves (at least) equal
> consideration in the standard as the sup-inf form.
>
> Remark 1. Narrow intervals indeed have certain important properties.
>
> Remark 2. Centered multiplication may be optionally implemented,
> whereas the "true" interval multiplication should be (of course) obligatory
> in the standard. Centered multiplication is only an approximation of
> the true one, but is faster.
>
> Svetoslav
>
>
--
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R. Baker Kearfott, rbk@xxxxxxxxxxxxx (337) 482-5346 (fax)
(337) 482-5270 (work) (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
Box 4-1010, Lafayette, LA 70504-1010, USA
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