RE: Motion P1788/M0037.01:MidAndRadSpecs: voting period restarted (unless objection)
I vote YES
-----Original Message-----
From: stds-1788@xxxxxxxx [mailto:stds-1788@xxxxxxxx] On Behalf Of Ralph Baker Kearfott
Sent: Monday, October 22, 2012 5:46 AM
To: John Pryce
Cc: stds-1788; Ralph Baker Kearfott
Subject: Motion P1788/M0037.01:MidAndRadSpecs: voting period restarted (unless objection)
P-1788:
Unless I receive objections, let us amend the motion to read Rad(Empty) = NaN, and immediately re-start the voting. The new voting period will end after Monday, November 12. I append the amended motion. (Those few who have already voted, please re-post a vote to the amended motion.)
Juergen: Please update the web page with this information.
Best regards,
Baker
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This motion about midpoint and radius is based on the discussions during our 2012 annual meeting at SCAN'2012, specifically on the idea proposed by Siegfried Rump:
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Definition of the _midpoint_ of an interval [a,b]:
* we compute the mathematical midpoint
(a + b) / 2 in the extended real line (whenever it is possible), and then take a finite computer representable floating point number which is the closest to this mathematical midpoint; if there are two closest numbers, we use rounding to even, i.e., select the one whose binary expansion ends in 0
* the only interval for which the mathematical midpoint is not defined is the interval (-oo, +oo); for this interval, natural symmetry prompts us to define the midpoint as 0;
Examples:
* for an interval [a, +oo) with finite a, the midpoint is the number closest to +oo, i.e., MAXREAL
* for an interval (-oo, a) with finite a, the midpoint is the number closest to -oo, i.e., -MAXREAL
* for an interval [1, 1 + u], where 1 + u is the number closest to 1, the mathematical midpoint is 1 + (u / 2), so the closest numbers are 1 and 1 + u; rounding to even results in 1 being the desired midpoint
* mid(Empty) is defined to be NaN.
Please note that a midpoint is, in general, different from a bisection point used to bisect an interval in different interval algorithms
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For any interval [a, b], once its midpoint m is defined, we can define its _radius_ r as the smallest computer representable floating point number (finite or infinite) for which the interval [m - r, m + r] contains the original interval [a, b].
Examples:
* for the interval [1, 1 + u], the radius is u
* for the intervals [a, + oo) and (-oo, a), the radius is oo; this example shows the need for using an infinite number.
* rad(Empty) is defined to be NaN.
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On 10/22/2012 06:13 AM, John Pryce wrote:
> Baker
>
> It looks as though there is a consensus, will you state the revised motion and let voting proceed?
>
> On 21 Oct 2012, at 23:56, Kreinovich, Vladik wrote:
>> Rad(Empty) = NaN is good too, let us go with Michel's suggestion
>
> I have another query, though it is about wid() so it doesn't affect this motion.
>
> I thought that with the revised definition that (for nonempty xx = [xlo,xhi])
> wid_F(xx) = smallest F-number >= xhi-xlo one would always have
> (*) wid_F(xx) <= 2*rad_F(xx).
> But this is not so as shown by the example xx = [-.001,1.00] where F is 3-digit decimal. It gives rad=0.501, wid=1.01.
>
> So though always wid=2*rad in exact arithmetic, both < and > can happen in finite precision.
>
> Are we happy with this, or should we, say, adjust the definition of rad_F so that (*) always holds?
>
> John Pryce
>
--
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Ralph 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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