Ulrich Kulisch wrote:
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.
But all this can be done as efficiently with the optimally rounded dot
product, since all that matters is that sufficiently many significant
digits are produced.
This is the reason, I think, why Siegfried Rump got interested in
algorithms for the latter. In any case, I'd like to see his comment
on this issue.
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.
Rump published work on how to get least significant bit accuracy for
arbitrary arithmetic expressions