MidRad and reproducibility
Vincent Lefevre wrote:
On 2009-09-21 13:22:33 +0200, Arnold Neumaier wrote:
Vincent Lefevre wrote:
There's no need to have a canonical definition or fix the algorithm.
The most important point is to have some properties that the result
satisfies.
The enclosing property need not be standardized since it is obvious
that any interval type must preserve this.
Well, not so obvious as some interval arithmetic packages don't
guarantee this property. An example in Maple + evalr. IMHO such
implementations must not be called interval arithmetic, but one
can't stop people from using wrong terms unless there's a standard
that makes everything clear.
One can still call an implementation interval arithmetic if one does
not claim to conform to the standard. A user needs to read the claims
made, whether the claim is for rigorous enclosure or for conformance
to a standard.
Any two different algorithms for computing an enclosure of the
midrad product of two midrad intervals are likely to give slightly
different results. Thus standardizing more is difficult without
tying down the algorithm.
I recall that I do not seek to require reproducible results.
It is not clear that infsup will require that either anyway.
The Vienna Proposal implies reproducibility for any computation
using only operations that are declared as tight.
If allowance is made for extra-tight interval operations using internal
higher precision, this would be another data type, and again conformance
to the standard would imply reproducibility for such results.
Of course, reproducibility can m,ean only reproducibility under the same
parameter settings (i..e, same data types for storage and for
intermediate results).
Can you please give an example of what beyond enclosure you want to
require from an implementation of midrad multiplication?
But it is not true that IEEE 754 does not standardize multiprecision.
It doesn't mention multiprecision explicitly, but it doesn't exclude
it either. Multiprecision can be seen as supported via the section
on the extended and extendable precisions (the binary representation
is left to the implementation). And this is better than nothing.
So, why shouldn't something similar be done for interval arithmetic?
We have that in Motion 6. And by not explicitly mentioning midrad in
the standard, we do the same as does IEEE 754 with multiprecision.
Arnold Neumaier