Re: A proposal for motion 5 "Arithmetic operations for intervals"
On 2009-06-04 12:32:38 +0200, Arnold Neumaier wrote:
> Ulrich Kulisch schrieb:
> >The notation [x, y) for an interval where the bound y is not an
> >element is quite usual in mathemeics. The notation [x, y] may
> >occasionally cause confusion.
>
> I can't see any possible confusiion in the interpretation of the
> interval (i.e., closed subset of the reals, as we already agreed)
> [0,Inf]. It can only mean the set of all real nonnegative numbers.
>
> However, there is considerable confusion if one insists on writing
> this interval as [0,Inf). For it means that I can never refer to an
> interval with lower bound zero and arbitrary upper bound u without
> a clumsy statement
> ''[0,u] if u is finite, and [0,Inf) otherwise''.
> It _must_ be possible to write an arbitrary interval with lower
> bound 0 as [0,u], no matter whether u is finite or Inf, and
> subsequent specialization to u=inf should not require a change
> in notation.
I agree.
> >>5. Arithmetic operations like unary minus, abs, square, sqrt, etc.
> >>are not covered. But some issues like the treatment of undefined
> >>values of the real operation affect these as well, and hence must
> >>be decided _before_ fixing what happens to special cases.
> >As the title of the proposal says it just considers arithmetic
> >operations for intervals.
>
> The term is not unambiguous. For example, Wikipedia says,
> ''The traditional arithmetic operations are addition, subtraction,
> multiplication and division, although more advanced operations (such
> as manipulations of percentages, square root, exponentiation, and
> logarithmic functions) are also sometimes included in this
> subject.''
> (http://en.wikipedia.org/wiki/Arithmetic_operations)
And nowadays, fma could be regarded as an arithmetic operation or not.
> Independently of that, my argument was that some issues like the
> treatment of undefined values of the real operation affect also
> elementary functions, and hence must be decided _before_ (or
> simultaneously with) fixing what happens to division.
Yes.
> >>6. The set of floating-point numbers cannot be regarded as a subset
> >>of R, since +0 and -0 are distinct floating-point numbers.
> >>Thus already the formal setting (line 3 of the proposal) is faulty.
> >Interval arithmetic and floating-point arithmetic are different
> >topics. In interval arithmetic -0, +0, and NaN should not be
> >considered to be floating-point numbers.
>
> In interval arithmetic,
> - either one does not need to consider rounding issues, but works with
> exact hulls;
> - or one needs to address the effects of representation by implementable
> floating-point data, and then has to consider the floating-point data
> actually available.
But floating-point data could be interpreted in different ways.
For instance, we can (and IMHO should) choose that -0 and +0 (in
Level 2 of IEEE 754) represent the same value corresponding to
the real value 0, and the sign of the representation would not
necessarily be specified. Said otherwise, we can consider a
subset of R (and the corresponding intervals). The other issues
are implementation issues.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.org/>
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Work: CR INRIA - computer arithmetic / Arenaire project (LIP, ENS-Lyon)