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Re: Motion P1788.1/M001.02 -- YES

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Dear Michel,

On 12/07/2015 10:08 PM, Michel Hack wrote:
> I vote YES to the text of Clause 4 of Nov 30, 2015.
> As for "Natural Interval Extension" -- it seems to me that this is
> the proper term.  Is is also consistent with the parent standard
> 1788-2015.
> Now, there may indeed be confusion if others have used this term
> to denote what I might call "Moore Interval Extension".
> So I would suggest, as an editorial comment, that we add the
> following at the end of Clause 4.4.4:
> The term "Natural Interval Extension" has frequently been used to 
> denote the result of expression evaluation, as opposed to the hull 
> of the range.  The term "Moore Interval Extension" might have been 
> more appropriate for this.
> Michel.
> P.S.  The Wikipedia article does not define the "function rule"
> that it applies to its definition of "natural interval extension". 
> The problem of course is that many expressions can be used to 
> compute a function, and some functions might not even use an 
> expression -- they might use table-based interpolation for
> example. So while there may be a "natural interval expression
> extension", the notion does not make sense for a *function*.
> An article by Alexandre Goldsztein uses "Natural A-E extension" 
> which however restrics the expression to single-use variables. 
> ---Sent: 2015-12-07 21:33:52 UTC

Two months ago, I searched my library for a definition of "Natural
interval extension". Here is what I found:

Introduction to Interval Analysis, Ramon Moore, R. Baker Kearfott and
Michael J. Cloud. SIAM 2009.
The authors define "natural interval extension" on Page 47 in an
informal manner (replace all real quantities and operators by interval
ones) that corresponds to my own understanding.

Applied Interval Analysis. Luc Jaulin, Michel Kieffer, Olivier Didrit
and Eric Walter. Springer 2001.
The authors define a "natural inclusion function" on Page 30, again by
mapping real quantities to interval ones. Still rather informal.

Validated Numerics. Warwick Tucker. Princeton University Press 2011.
Warwick defines a "natural interval extension" informally as
substitution from real objects to interval ones.

I could not find the expression in "Interval Methods for  Systems of
Equations" by Arnold Neumaier. However, I surmise he would sanction
the syntactical definition, as it is the one used in a paper he
coauthored with Daney and Papegay:
Interval Methods for Certification of the Kinematic Calibration of
Parallel Robots. D. Daney, Y. Papegay, and A. Neumaier. Proceedings of
the 2004 IEEE International Conference on Robotics & Automation.

I believe there is enough papers and books that do not use "natural
interval extension" in the sense used in P1788.1 to make its
(re-)definition there objectionable.

Best regards,

- -- 
Frédéric Goualard                                 LINA - UMR CNRS 6241
Tel.: +33 2 76 64 50 12    Univ. of Nantes - Ecole des Mines de Nantes
                                   2, rue de la Houssinière - BP 92208                   F-44322 NANTES CEDEX 3

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