Re: the "set paradigm" is harmful
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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