Re: "natural interval extension"
P1788
I agree with Markus, in that Frédéric is talking about Level 2, but we are discussing the Level 1 definition/terminology.
So the effect of finite precision is irrelevant. I think the relation between Levels 1 and 2 is correctly made in the standard; at least people such as Michel H and Vincent L examined it carefully, proposed many changes, and approved the final version.
However Markus: with respect, "impose conventions like x+y*z meaning x+(y*z)" is a language issue, and not about either Level 1 or Level 2.
I would be happy if the only action we take now is to add the note Ned proposes. Longer term, maybe change "natural interval extension" to "range-hull extension" or another name.
Maybe "functional interval extension"? My case for that is that the end-purpose of interval computation is not to prove things about expressions but to prove things about real (or complex) functions. The expressions are *always* the servants of the functions. We enclose the solution of a linear system, or an eigenvalue of a PDE, or prove Kepler's conjecture.
So I agree with Markus's saying
On 8 Dec 2015, at 17:36, Neher, Markus (IANM) wrote:
> 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).
but not quite with the conclusions he draws from it.
Regards
John
On 8 Dec 2015, at 21:59, Neher, Markus (IANM) [IANM ist die Organisationseinheit Institut für Angewandte und Numerische Mathematik am KIT] <markus.neher@xxxxxxx> 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...
>
> 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?