Re: The current proposal
Siegfried M. Rump schrieb:
On Wed, 25 Feb 2009 15:41:02 -0100, Arnold Neumaier
<Arnold.Neumaier@xxxxxxxxxxxx> wrote:
Always???
This value is not even properly defined in many cases,
such as when F(x)=x-x or F(x)=(x-1)/(x+1) and A=[0,inf].
This was spoken in terms of established practice in the
mathematical community at large. I find it extremely important
that we do not create sectarian deviations from very established
practices.
Yes.
We can change with the standard the practice of interval computation,
but not the practice of mathematics.
What about F(x)=(x^2-1)/(x-1) for x=1? There seem some choices to
implement this function following the Vienna proposal, but if I
understand it correctly, the result is always Empty.
Given this, does this change practice in mathematics?
Standard practice in mathematics evaluates F(x) at x=1 to 0/0
and concludes that F is undefined at x=0. On the other hand,
f(x):= 2 if x=1, f(x):=(x^2-1)/(x-1) otherwise
defines an everywhere continuous function. The interval evaluation
of f(x) is defined by
f(A)= (A^2-)/(A-1) if 1 not in A,
f(A)= convexHull(2, (A^2-)/(A-1)) otherwise,
and produces f([1,1])=[2,2], as desired.
In mathematics there are exactly two possibilities, either
the result F(1) is undefined or, after some obvious modification,
the result is f(1)=2. It is not the empty set.
F(1) is undefined (NaN), and F({1,1]) is the empty set, as it should be.
An interval enclosure is the hull of all results where the function
is defined. The hull of an empty set of results is the empty set.
Similarly, f(1)=2, and f([1,1])=[2,2], as it should be.
Similarly, for g(x)=sqrt(x), g(-1) is undefined, and g([-1,-1])=Empty.
If users specifies exactly which function they mean, they get a
valid result. If they don't, they need not be surprised to get
unexpected results.
Is it reasonable that a mathematically defined object, which
may be the result of legal operations such as intersection,
is used to indicate something else, is used as a flag?
Can you give an example of what you mean here?
For f(x):=(x^2-1)/(x-1), f(1) is not mathematically defined.
Arnold Neumaier