Re: Midpoint and unbounded intervals
The property (22) doesn't always hold.
Let us cosider two-digit floating-point decimal numbers.
X = [1.0,1.1]
midpoint(X) = roundNear((1.0 + 1.1)/2) = roundNear(1.05) = 1.0
The definition (24) says
radius(X) = roundUp((1.1 - 1.0)/2) = roundUp(0.05) = 0.05
The interval [midpoint(X)-radius(X),midpoint(X)+radius(X)]=[0.95,1.05] doesn't contain X.
There tightest radius(X) satisfing (22) is:
radius([u,v]) = roundUp(max(midpoint([u,v]) - u, v - midpoint([u,v]))) .
However, it is necessary to explore if it can be implemented efficiently.
-Dima
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Отправленные: Понедельник, 30 Январь 2012 г 20:35:39 GMT +03:00 Москва, Санкт-Петербург, Волгоград
Тема: Midpoint and unbounded intervals
P1788,
Attached is an updated version of the midpoint paper. Many thanks to Michel,
Dmitry, Vincent and others who supplied comments and suggestions. I believe
we've incorporated all of them.
Sincerely,
Nate & Dan