Re: Two kinds of interval arithmetic domains
Dear Michel,
I stronly support the view that there are two main
application domains of IA, and both domains should
be taken into account in the standard. I append below
my posting of 09 Feb 2009 entitled the
"set paradigm is harmful", where I objected the view
that the standard should concentrate only on
(in your terminology) "the second domain -- that
of containing results that depend on parameters within
certain ranges". In later posting I pointed out the
"settable uncertainty threshold" for narrowness of
intervals from "the first domain": Rad <= 0.4 |Mid|.
To the last paragraph in your posting I would
add that, vice versa, technics from the first domain can
be used (by experts) to derive algorithms to help
users in the second domain (which ought to be
accessible to beginners). For example, one can use both
endpoints of a bounded set-interval as approximate
numbers in order to get simultaneously outer and inner
bounds (so-called twin IA).
In addition, as I mentioned in my posting of 09 Feb 2009,
both domains should be extended to complete
(Kaucher/modal) interval arithmetic, which can be invisible
for naive users.
Best regards,
Svetoslav
On 19 Apr 2009 at 11:51, Michel Hack wrote:
> Reading these last few discussions on whether an IA standard should
> address experts or naive users (presumably both, though perhaps in
> different ways), it occurred to me that there are at least two rather
> different ways that IA might be used, and that the intuitions in the
> two cases may be very different.
>
> The first domain is that of reliably-bounded computations with uncertain
> numbers. In this domain all intervals are expected to be bounded and
> relatively narrow. If intermediate interval results become unbounded,
> or perhaps even exceed a settable uncertainty threshold, it might be
> useful to stop the computation and look out for a different approach:
> use a different algorithm, retry with higher precision, or give up and
> report an ill-conditioned problem. This is the domain where MidRad
> representations may have many advantages, and where the precision of
> the bounds loses its significance (pun intended) as intervals become
> too wide to be useful.
>
> The second domain is that of containing results that depend on parameters
> within certain ranges. Any given parameter value may be highly precise,
> but is picked from a bounded (or semi-bounded) range. This is the domain
> or forwards and backwards containment evaluations, and of the various
> flavours of "non-standard" intervals. Here the precision of the bounds
> can be critical, e.g. in the vicinity of singularities -- sometimes it
> even matters whether the bounds themselves (as FP numbers) are considered
> to be included or excluded.
>
> The two have much in common of course -- in particular, the techniques of
> the second domain can be used (by experts) to derive algorithms to help
> users in the first domain (which ought to be accessible to beginners).
>
> Any thoughts on this aspect of things?
>
> Michel.
> ---Sent: 2009-04-19 16:11:55 UTC
======================================
To: stds-1788@xxxxxxxxxxxxxxxxx
Subject: the "set paradigm" is harmful
Date sent: Mon, 09 Feb 2009 09:54:50 +0200
Dear colleagues,
The phase:
(S) "intervals are sets of numbers",
is repeatedly stated in the documents
discussed by now.
Thereby (S) is understood in the sense,
that intervals are boxes of the form [a, b].
I shall further call this theoretical framework
the "set paradigm".
In my opinion a standard based on the set paradigm will
be able to serve only a limited number of applications and
is harmful for the future development of interval analysis.
The set paradigm excludes the view the intervals can be
considered as approximate numbers. (S) imposes the
priority of the inf-sup presentations of intervals and thus
prohibits the equally important view that intervals
are approximate numbers (having a "main" or "mean"
value and error bound). It has been shown that intervals
presented in midpoint-radius notation satisfy the same
algebraic structures as intervals using inf-sup presentation.
The argument of some supporters of the set-paradigm
that an interval of the form [a, oo) cannot be presented in
midpoint-raduis form seems to me ridiculous.
The set-paradigm implies the consideration of only
the "standard" interval arithmetic structures and prohibits
their natural algebraic extensions. The supporters of (S)
argue that every statement using improper intervals
can be presented by proper ones. Imagine that in
the IEEE 754 standard one adopts the thesis that negative
numbers are excluded of consideration. Indeed, every
proposition using negative numbers can be modelled by using
just positive numbers! To those who know the embedding theorem
(for the embedding of a semigroup into a group) thesis (S) is
equivalent to prohibiting negative numbers.
The set-paradigm leads to unnecessary limitations in the
application of the standard.
Interval arithmetic structures should be considered in the way
we consider real arithmetic structures. Interval arithmetic structures are
rich structures arising from just two basic operations and one relation.
These abstract structures can be used for various purposes depending on
the semantic contents which one attaches to the models/problems of
consideration. It is true that the idea of a set is underlying in the
semantic contents but with various nuances. Algebraically complete
intervals structure are needed when using advanced techniques of interval
analysis.
The extraordinary diversity of the extended interval arithmetic structures
allow various uses and interpretations, which incorporate
the set paradigm as a special case. This paradigm in its narrow sense is
harmful for the future development of interval analysis.
The implementation of complete interval arithmetic does not require
additional costs. Apart from standard applications it can serve for many
other purposes in extended applications such as modal interval analysis,
range interval analysis using inner interval arithmetic operations etc.
Up to certain extent Kahan intervals can be also modelled by generalized
intervals.
S. Markov
-