Motion P1788/M0037.01:MidAndRadSpecs -- discussion period begins
P-1788:
Since Motion 37 has been made by Vladik Kreinovich and
seconded by Nate Hayes, the discussion period now
begins, and will end after Friday, October 19, 2012.
I append the motion.
Discussion on this motion will proceed according to the rules for
position papers.
Juergen: Please place the motion and associated information
in the appropriate place on the web page, as
you have aptly done in the past. (Of course,
we expect you do so after you have safely completed
your travel.)
Nate and Vincent: Thank you for starting the discussion.
Acting secretary: Please record the transaction in the minutes.
As usual, please contact me if you need the password to the private
area of the P-1788 web site.
Best regards,
Ralph Baker Kearfott (acting as chair, P-1788)
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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
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.
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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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