Motion9 ExactDot Product

[Ulrich Kulisch <Ulrich.Kulisch@xxxxxxxxxxx>]

Arnold Neumaier wrote:

> > I had therefore asked for providing evidence for applications that really
> > need the exact dot product, but this hasn't generated any response.
>   
There was some response: I copy a few sentences of my response mail (dated October 20):

The IFIP Working Group letter requests an "exact" dot product as support for long interval arithmetic (for details see [6] or [9]). Long interval arihtmetic opens a new area of 
applications for interval arithmetic. I give a few examples:

The Krawczyk operator frequently is used to compute a verified enclosure 
for the solution of a system of linear equations. But for ill 
conditioned problems (take the Hilbert matrices of dimension larger than 
11) the Krawczyk operator fails to find the solution.

In such cases a more powerful operator (called the Rump operator in [9]) 
solves the problem. In its extended form the Rump operator can be used 
to compute a verified solution even for extemely ill conditioned 
problems, see [11]. The Rump operator uses the "exact" dot product.

Another very nice example is given in [7], an iteration with the 
logistic equation. Double precision floaint-point or interval arithmetic 
totally fail (no correct digit) after 30 iterations while long interval 
arithmetic after 2790 iterations still computes correct digits of a 
guaranteed enclosure. Long interval arithmetic allows computing highly 
accurate bounds for real arithmetic expressions.

Best regards
Ulrich Kulisch