Re: Siegfrieds recent paper
On 2012-01-05 15:33:13 +0100, Siegfried M. Rump wrote:
> Agreed. I do not suggest half-open or open intervals. I introduced
> a certain symmetry between overflow and underflow. Basically
> 1/inf gives T and 1/T gives inf, in a mathematically sound way.
Well, inf is used for two different things here: for an infinite bound
(in the usual mathematical way) and for a finite bound when an overflow
occurs. If you want a real symmetry, you'll have to introduce the
"inverse" of T (in addition to the usual mathematical inf).
> > If you do not assume incorrect conclusions, there would not be any
> > problem.
>
> Correct. But definitions should be as people expect.
Not all people expect the same things.
> > In a similar way, you shouldn't assume that every element
> > of the returned interval is in the image of the function.
>
> However, this is the usual way of thinking of people using interval
> arithmetic, interval arithmetic has been used this way for years.
> If the result is meaningful (i.e. not NaI or alike), then the
> underlying function is well-defined and the range enclosed.
I would say that people took bad habits.
If f is defined on [-pi,pi], this makes things like computing an
enclosure of simple expressions like f([0,pi]) difficult.
> In many cases functions are well-defined. Since users are not used to
> check flags to ensure that a result is correct, it may be forgotten.
A standard should be written for the future, not for current users.
Implementers could provide compatibility wrappers for them.
> For example, in measure theory people are used to define inf-inf=0. In
> IEEE754 the choice was inf-inf=NaN. One could also have defined
> inf-inf=0 with a flag; but the majority of numerical analysts would
> prefer (or expect) inf-inf=NaN.
IEEE 754 is not for measure theory. The IEEE-754 inf is first a
consequence of overflow, not the inf of measure theory. In this
context of floating-point, NaN was the right choice for inf-inf.
> Agreed. I propose to define as default what the majority expects. And I
> think occasional as well as expert users of interval arithmetic would
> expect that a proper result interval implies a well-defined function.
>
> IMHO a good way is to define an expert mode including decorations. For
> the occasional and even for many expert users decorations are a
> dangerous trap.
I disagree. Interval arithmetic should be used to get answers that
can be used (with more information than NaI), not to detect bugs
in users' programs.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / Arénaire project (LIP, ENS-Lyon)