Re: modal interval
Kreinovich, Vladik schrieb:
I agree with Alexandre that we seem to be not yet ready to include modal into the proposed standard.
[...]
* to Nate modal interval techniques mean a very specific (and, as I understand, very efficient) methodology implemented in his software.
I do not think anyone is well familiar with that methdology, and many of us are eager to learn.
So, when Arnold is arguing against specific features of modal intervals, he means modal intervals in the general sense.
When Nate argues that Arnold does not know modal intervals, what he means is that Arnold does not know his algorithm well
(which of course is true).
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.
I know and understand all of Nate Hayes manuscripts and patents on
modal intervals that are publicly available, and I know and understand
probably more of the other literature on nonstandard intervals than
Nate Hayes.
Nothing there allows the conclusion that, for range enclosures,
modal intervals can achieve more than traditional monotonicity
analysis together with directed rounding and standard interval
techniqies.
Modal techniques can possibly evaluate equivalent range enclosures
in a slightly more automatic or faster way, given appropriate hardware
support.
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.
I believe, however, and argued on p.27 of my survey paper
Computer graphics, linear interpolation, and nonstandard intervals
http://www.mat.univie.ac.at/~neum/ms/nonstandard.pdf
why, that with similar effort in the hardware implementations,
one can do with ordinary interval arithmetic everything that
modal arithmetic can do for range enclosures, and with similar
efficiency.
Thus I do not believe that modal/Kaucher interval arithmetic give
any significant advantage for range enclosures. At least, none has
been demonstrated in the existing literature.
Since any justification for the standard to be must be based on
public evidence, this is enough to justify non-inclusion of
modal/Kaucher interval arithmetic into the standard.
Kaucher intervals are, however, more widely applicable than just
to range enclosures, and may be useful for a number of applications
involving tolerance problems; see Shary's and Popova's work quoted
in the survey.
Therefore I advocate (in the Vienna Proposal) that nonstandard
intervals are provided with minimal support, so that users of
Kaucher arithmetic are not hampered by the standard.
Arnold Neumaier