Re: Balance
Ulrich Kulisch schrieb:
Dear George and others,
thank you for your mail. This is a mail I really enjoy! An interval
arithmetic standard should be as simple as possible.
The plenty of mail I am getting every day exceeds my capacity. So I
cannot comment on everything as I perhaps should. Sorry for that.
Let me first comment on a few minor things.
1. I remember several mails where simultaneously 'inf' is used as an
abbreviation for infinity as well as for infimum (the greatest lower
bound). Can't we avoid this? infty or +oo would be better abbreviations
for infinity.
In the next version of my proposal, I'll use Inf for infinity and inf()
for infimum.
2. Frequently intervals like [-inf, a], [b, +inf] or [-inf, -inf],
[+inf, +inf] appear, immediately followed by the sentence that "the
bounds -inf or +inf are not elements of the set". I think in such cases
it would be much better to use parantheses instead of brackets. This is
fully consistent with IEEE P754. The intervals then would be written as:
(-oo, a], [b, +oo), or (-oo, -oo), (+oo, +oo) respectively. The latter
two notations would naturally indicate that the set is empty.
The problem is that then every time an interval with arbitrary bounds
is mentioned, one needs to distinguish four cases. Even simple rules like
[l,u]+[l',u']=[l+l',u+u']
cannot be formulated without exceptions if we do not allow writing
[-inf,a], [b,+inf] or [-inf,-inf].
3.Formulations like:
"NaN, -inf, +inf, and 0 denote any floating-point number x such that
value(x) is undefined, -inf, +inf, and 0, respectively; it is assumed
that such floating-point numbers exist. inf and +inf are used
synomymously."
should be avoided. NaN is not a number and consequently not a
floating-point number. IEEE P754 distinguishes between floating-point
numbers and floating-point data. Every floating-point number is a real
number!
No. According to IEEE 754-2008, every floating-point number is a real
number or +-Inf.
In the next version of my proposal, I'll use the term numeral to stand
for any of integers, finite floating-point numbers, fixed-point numbers,
Inf, -Inf, ir NaN.
4. Can anybody tell me whether the Hausdorff metric can be defined for
extended intervals?
It is defined for two arbitrary sets in a metric space, hence for
intervals, closed and connected sets of reals.
Note that
set round up
distance([l,u],[l',u']) = max(abs(l-l'),abs(u-u'))
provided that inf-inf is zero under directed rounding (which is
currently not the case but is desirable, also for various other
formulas)
The same definition is meaningful for nonstandard intervals.
Arnold Neumaier