Re: "natural interval extension"
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Dear Markus,
On 12/08/2015 06:36 PM, Neher, Markus (IANM) [IANM bezeichnet die
Organisationseinheit Institut für Angewandte und Numerische
Mathematik] wrote:
> 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.
Au contraire, this is the typical definition found in most books, and
it is severely lacking in several aspects as soon as you need
computing with intervals. Nothing is said about the replacement of the
operators, included the 0-ary ones---aka constants. Consider the
expression:
f(x) = x + 0.1
For all *practical* purposes, you cannot merely replace the
real-valued variable by an interval-valued one to get the "natural
interval extension" of f(x). You also would have to replace the
addition by its interval extension (definition?), and the constant
"0.1" by the smallest interval---in the sense of the inclusion---that
contains "0.1". Ratschek and Rokne's definition is not good enough to
specify all this. As are many (most) other definitions in the literature
.
Best regards,
FG.
- --
Frédéric Goualard LINA - UMR CNRS 6241
Tel.: +33 2 76 64 50 12 Univ. of Nantes - Ecole des Mines de Nantes
2, rue de la Houssinière - BP 92208
http://frederic.goualard.net/ F-44322 NANTES CEDEX 3
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