Re: P1788: Our first formal motion has entered its discussion period
Ulrich Kulisch schrieb:
Arnold Neumaier schrieb:
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1. Ulrich Kulisch wrote:
Concerning StandardizedIntervalNotation I would like to make two
remarks:
(in this mail R stand for the set of real numbers and I for an
interval set.)
1. Of course basic to all considerations is the set of extended reals
R*.
Interval arithmetic deals with closed and connected sets of real
numbers (and nothing else). If an interval is bounded it is written
as [a, b] with a, b elements of R. If it is unbounded it is written
as (-oo, a] or [b, +oo) with a, b elements of R or (-oo, +oo) where
the parantheses indicate that the bounds -oo and +oo are not elements
of the interval.
How do you write an interval with bounds a and b if you don't know
whether or not a and/or b are finite? This is a very common situation.
The Vienna Proposal writes (like the much-used Intlab) in any case
[a,b], and thus dispenses with the need for two kinds of brackets.
The set of all such intervals should be denoted by IR. Then
{IR, +, -, *, /} is an exception free calculus.
So what do the proposed notations like IR* or *IR^n really mean? If
we really consider intervals of IR* we would have to allow intervals
like [-oo,-oo] and [+oo, +oo].
These are in IR^* but not in IR. Standard interval arithmetic is for IR;
so there is no conflict here.
I cite from the proposed StandardNotation:
A (real, closed, nonempty) interval is a 1-dimensional box,
i.e., a pair x = [\ul x, \ol x] consisting of two real numbers
\ul x and \ol x with \ul x <= \ol x. The set of all
intervals is denoted by IR.
In my understanding this definition excludes the empty set and intervals
like (-oo, a] or [b, +oo) with a, b elements of R or (-oo, +oo) from IR.
In my definition above these are included in IR.
In StandardNotation, they are in *IR, the set of ''extended intervals''
defined on p.8. These are called ''textbook intervals'' in the Vienna
Proposal. A ''standard interval'' in the Vienna Proposal is an element
of *IR whose endpoints are B-numerals.
StandardNotation was devised for traditional interval arithmetic,
where intervals were closed, bounded and nonempty (as in the books
by Moore, Alefeld/Herzberger, or myself). StandardNotation codified
the actual practice in publications. Publications using unbounded
intervals are much less frequent, so that the more complex notation
*IR, adopted in StandardNotation, seemed justified.
In my opinion, changing the traditional meaning of IR would be
confusing. For maximal clarity, the Vienna Proposal avoids both IR
and *IR.
Arnold Neumaier