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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