Re: "natural interval extension"
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Dear Markus,
On 12/08/2015 10:59 PM, Neher, Markus (IANM) [IANM ist die
Organisationseinheit Institut für Angewandte und Numerische Mathematik
am KIT] wrote:
> Dear Frédéric,
>>> 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
>
> In "real" interval arithmetic (intervals with real bounds, no
> roundoff), 0.1 and + are well defined. The same holds for rational
> expressions. For roots or transcendental functions, f(X) =
> range(f,X) if X \subset domain(f) can be used (I'm aware that there
> are alternatives to this).
>
> For "machine" interval arithmetic, you have to specify the basic
> arithmetic operations, transfer functions such as x \in R to X \in
> IR, and the elementary functions *first*. You also have to impose
> conventions like x+y*z meaning x+(y*z) for interval operations,
> that is X+Y*Z meaning X+(Y*Z). Nevertheless, once these atoms are
> defined, the definition of Ratschek and Rokne applies to
> expressions. Am I missing something?
An interval extension of a *function* is defined in terms of
properties it must have. As said copiously before, a natural interval
extension is related to an *expression* and not a function. It is
about covering an expression tree with interval-valued variables and
interval operators in a manner that is deemed "natural". As Svetoslav
said yesterday, "natural" is a relative notion. If I have, say, a
specialized interval operator "+(X,Y,Z)" that computes a narrower
interval for X+Y+Z than by using the addition twice, I can cover, e.g.
the expression
3*(A+B+C*D)
in more than one way, which will give different results.
Which cover is the "natural" one is not explicitly specified by the
definition you quote.
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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