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Re: MidRad -- and Two Different Application Domains (TDAD).

Vincent Lefevre wrote:

On 2009-09-21 10:46:39 +0200, Arnold Neumaier wrote:

There is not even a canonical definition of what it could mean to
perform a midrad multiplication in finite precision arithmetic.
Because of rounding error issues, it would be very difficult to
standardize it without fixing the algorithm that computes thing.


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.

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. 



Also note that even if you have a canonical definition for the infsup
multiplication, this doesn't mean that you'll get the same result on
every platform. For instance, the underlying arithmetic can still have
extended precision. And things will get more complicated for the
elementary functions. And what about expressions like A + B + C?
Do you have a canonical definition that would satisfy all the users?
For instance, if A = x and C = -x (x as a variable), would you allow
an implementation to optimize this expression to B?


Each multiple precision package should handle it in its own way....


This is precisely what I want to avoid. Why should there be a standard
for small fixed precision, but not for multiple precision?
For the same reasons as IEEE 758 did not standardize multiprecision arithmetic. One needs to limit what is standardized since the capacity 
of any committee is limited. Less is more.


Arnold Neumaier