I think we already lost the battle with respect to this motion, and I
think people do not want to hold it hostage to a reasonably minor
thing, especially since we have the same unfortunate wording in 1788.
So maybe someone can formulate a new motion proposing to replace this
confusing term with someone more appropriate term, whether it is
convex hull of the range or any other term. Before we make this
motion, we can discuss which term to use.
There are quite a few of us who are strongly against using this
confusing term, and I think everyone agrees that the term is
confusing, so hopefully our new motion -- once we agree on the
appropriate term -- will pass.
________________________________________
From:stds-1788@xxxxxxxx <stds-1788@xxxxxxxx> on behalf of Neher,
Markus (IANM) [IANM bezeichnet die Organisationseinheit Institut für
Angewandte und Numerische Mathematik]<markus.neher@xxxxxxx>
Sent: Tuesday, December 8, 2015 10:36 AM
To: John Pryce;<stds-1788@xxxxxxxxxxxxxxxxx>
Subject: Re: "natural interval extension"
John et al,
On 08.12.2015 13:25, John Pryce wrote:
A.If one thinks of "interval extension" as being a property of a
*function*,
I maintain that the definition in 1788 is the most "natural
In my opinion, a function has no need for an interval extension. If you
want to refer to its range, call it its range (BTW: in several
dimensions, even for a continuous function, the range of f:X->R^n on
some interval X is not an interval in general). If you want to refer to
the (convex, interval, ...) hull of its range, call it the (convex,
interval, ...) hull of its range.
B.If one thinks of it as being a property of an*expression*, then
the one
that's "common in the literature" is sort of natural.
Computing range bounds of functions by evaluating*expressions* is one
of the basic tasks of interval arithmetic (and, in my opinion, also one
of its greatest accomplishments). While natural interval extensions are
redundant for functions, they are essential for expressions.
Or it would be, if
the books/articles making the definition made clear that many
different
expressions can define the same function. But they don't. Even
Warwick
Tucker's book, which I find exemplary in most ways, is vague, as
Michel
Hack pointed out on 7 Dec. (WT promised me to make it clearer in
a future
edition.)
The clearest definition I am aware of is given in the book by Ratschek
and Rokne, Computer Methods for the Range of Functions (1984):
If f(x) is an expression and X an interval of the same dimension as x,
then the (interval-)expression which is obtained by replacing each
occurrence of x in f(x) by X is denoted by f(X). The expression f(X) is
then called the natural interval extension of f(X) on X.
Three expressions, no functions involved.
Could we keep the main 1788 text unchanged but insert a footnote
about it as a corrigendum?
If we could, I'd strongly support this. We should not pursue a
misleading wording, neither in 1788.1 nor in 1788.
Regards,
Markus