RE: Motion 52: final "Expressions" text for vote
We have gone through these questions when we started the standard.
The range of a function over the interval is defined as the set of all possible values of the function when its arguments are in the range. If for some values within the range, the function is not defined, there are no values to add to the set.
For 1/x over [-1,+1], the actual range is (-oo,-1] union with [1,+oo), so the smallest interval containing this range is (-oo,+oo).
The function 1/(1/x) is equal to x when x is different from 0 and not defined for x =0. So, its range on the interval [-1,+1] is the same interval but without the point 0. The smallest interval containing this range is therefore the interval [-1,+1].
similarly, one can easily answer all your other questions.
What you describe is a very legitimate concern, that is why we have decorations, they enable us to distinguish between cases when the function is defined everywhere on the box and when it is not. John has a formal definition of decorations and an extension of the Main Theorem of Interval Computations to intervals wit decorations.
________________________________________
From: Bill Walster [billwalster@xxxxxxxxx] on behalf of G. William (Bill) Walster [bill@xxxxxxxxxxx]
Sent: Saturday, November 23, 2013 12:14 PM
To: Kreinovich, Vladik; Michel Hack; stds-1788
Subject: Re: Motion 52: final "Expressions" text for vote
Vladik,
The difficulty is that a given interval can contain values that are
outside the natural domain of definition of a given function or
operation. For example what is the smallest set of intervals that must
be included in the result of the following operations on the interval X
= [-1, +1]:
1/X
1/(1/(x))
ln(X)
1/ln(X)
X/X
X/X - X/X
The answers have nothing whatever to do with implementation or a
standard for it. The answers do depend on the definition of the
universal set of intervals and the real numbers contained therein and
the natural domain of any given expression and/or function. For example,
in addition to real numbers in the open interval (-oo, +oo), shall
intervals include {-oo, +oo} and/or oo = 1/0 and their reciprocals? If
so, how?
Shall there be an empty interval that is not the same as the empty set
because the empty interval is not a subset of every interval? If so,
when must the empty interval be included in the result of an interval
expression evaluation to prevent a containment failure?
Are there any conditions under which X - X = 0 for given non-degenerate
interval variables X?
Cheers,
Bill
On 11/22/13 8:04 PM, Kreinovich, Vladik wrote:
> For intervals proper, the defiition is the same always, f([x1],...,[xn]) is the smallest interval that contains the range of the given function f(x1,...,xn) on given intervals [x1], ..., [xn], i.e., the set of all the values
> {f(x1,...,xn):x1 is in [x1], ..., xn is in [xn]}.
>
> The questions arise when we want to extend this to situations when [xi] are not intervals
>
> (there are also definitions for decorations)
>
> ________________________________________
> From: Bill Walster [billwalster@xxxxxxxxx] on behalf of G. William (Bill) Walster [bill@xxxxxxxxxxx]
> Sent: Friday, November 22, 2013 2:31 PM
> To: Kreinovich, Vladik; Michel Hack; stds-1788
> Subject: Re: Motion 52: final "Expressions" text for vote
>
> Vladik,
>
> Then please point me to the purely mathematical foundation for computing
> with "well defined" intervals.
>
> Cheers,
>
> Bill