Re: The current proposal
Ralph Baker Kearfott schrieb:
Arnold et al,
Please see my inserted comments.
Baker
On 2/23/2009 11:22 AM, Arnold Neumaier wrote:
Siegfried M. Rump schrieb:
(1) For a given function F, for example, X=F(interval(A.sup)) should
give an
inclusion of the value of F at the right bound of A.
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 extermely important
that we do not create sectarian deviations from very established
practices.
We can change with the standard the practice of interval computation,
but not the practice of mathematics.
Have we already discredited Cset arithmetic?
I am voicing ''my'' advice, not the consensus of ''we'' = the 1788 group.
I discredited Cset arithmetic as much as I was able to.
It was a temporary fix to problems that surfaced in constraint
propagation and the interval Newton method.
This temporary fix created its own set of problems (unnecessary
overestimation in important cases for precisely the application
it was intended to cure).
In my assessment, it is a dead end.
The values
are clearly defined in that system, regardless of whether
or not we consider infinity to be a member of the set
of numbers.
According to the Reliable Computing 2006 document, Cset theory does
not define function values (which are numbers) but Csets, which are
sets.
For example, let F(x)=(x-1)/(x+1).
If A=[0,1] then
F(A.sup) = F(1) = 0
but the cset of F at A.sup is
F^*(A.sup)={0}.
And for A=[0,inf], the cset of F at A.sup is
F^*(A.sup) = F^*(inf) = {1},
while the function value F(A.sup) is undefined.
Arnold Neumaier