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.