2010/9/15 Arnold Neumaier
<Arnold.Neumaier@xxxxxxxxxxxx>
Arnold Neumaier wrote:
Rudnei Cunha wrote:
I strongly agree with Nate. I've seen enough evidence in the field of
numerical linear algebra using interval arithmetic - both standard and
mid-rad representations - that have convinced me that, in this field at
least, mid-rad is the best choice.
Could you please share with us enough references to that evidence
that support your conviction, so that others can check them for
their merits?
Mariana Luderitz Kolberg and coworkers. In particular, the most
detailed of these papers,
http://vecpar.fe.up.pt/2008/papers/19.pdf
mentions midrad arithmetic, referring to Rump's IntLab, which I had
already commented on.
In their algorithm for fast parallel interval system enclosures they
do not use midrad _arithmetic_ (although they claim so, erroneously)
but, as IntLab, real matrix-matrix and matrix-vector BLAS computations
with directed rounding, using the midpoint matrix and the radius matrix;
cf. their Algorithm 2.
Note that the mere occurrence of a midpoint matrix or radius in an
algorithm (which often occurs in applications) does not constitute
midrad arithmetic in the sense under discussion in the present standard.
Moreover, on p.7 they write that for most efficient performance
they _also_ use infsup intervals (apparently for the subsequent
iterative phase; details are not given).
In particular:
(i) Even algorithms based upon a midrad matrix representation need
infsup arithmetic for optimal performance.
(ii) A midrad-only implementation of a future P1788 standard
would not allow the implementation of the best performing version.
(iii) Standardization of midrad interval operations on a level
below BLAS2 would be of no use in fast interval linear algebra
algorithms.
Therefore nothing at all is lost for these applications if P1788
mostly ignores midrad interval arithmetic. It must be considered
only when (a new committee is) working on an interval BLAS standard.
Arnold Neumaier
For the only evidence I have seen where midrad is preferable
is vectorized matrix-vector and matrix-matrix multiplication
of matrices with narrow entries,
But an efficient implementation of that is completely independent
from having an implementation of midrad arithmetic operations, etc.
on the level of single operations.
Indeed, it is already efficiently implemented in IntLab using
directed rounding.
Since P1788 is not about standardizing interval BLAS but about lower level issues, a midrad implementation on the level of the standard
could not the slightest help in making vectorized matrix products
faster than the algorithm used by IntLab.