Re: modal interval
Nate Hayes wrote:
Arnold Neumaier wrote:
Modal techniques can possibly evaluate equivalent range enclosures
in a slightly more automatic or faster way, given appropriate hardware
support.
I'm glad to hear you admit that much, Arnold.
... with the stated qualifications;
- possibly,
- given appropriate hardware support,
- it would have to be demonstrated on a realistic benchmark
At the moment, it is only based on hearsay from you, without any support
by evidence. The evidence quoted below is spurious, since it beats only
a straw man.
Nate Hayes wrote:
> Arnold Neumaier wrote:
>> It is impossible to know undisclosed, nowhere detailed methods.
>> To argue with such confidential knowledge may be good marketing,
>> but is against the scientific spirit.
>>
>> ...
>>
>> But to claim this as a scientific fact, it would have to be
>> demonstrated on a realistic benchmark, or at least on a nontrivial,
>> realistic example.
>
> Chapter 6 of my paper gives one such example. It is simple example,
but not
> trivial and not insignificant, since Beziers, b-splines and NURBS form
> foundation of global industry in fields such as CAD, engineering, desktop
> publishing and computer graphics.
You compare it only with ordinary interval evaluation,
which is easy to beat. Compared to this, most other methods look good.
It is like praising a new optimization method because it is faster
than steepest descent.
But compare it against the use of monotonicity methods, as given in
my paper, and you'll find no advantage anymore. (I think, the
monotonicity methods had even a 2% advantage.)
Well, you say, monotonicity methods plus directed rounding are nothing
else than modal intervals in disguise, but this is in the eye of the
beholder.
With the same right one can say, modal intervals (as used for range
computations) are nothing but monotonicity methods plus directed
rounding in disguise, which would mean that modal intervals are
not needed at all since the traditional methods (monotonicity,
directed rounding) are sufficient to reproduce the modal results.
Even with more right - for to apply modal techniques, one needs
total monotonicity, while the endpoint method can work with simple
monotonicity. Thus the latter is of broader applicability.
Arnold Neumaier
Arnold Neumaier