Re: How do I bisect unbounded intervals?
On 2012-01-17 17:40:16 +0000, John Pryce wrote:
> On 17 Jan 2012, at 12:43, Vincent Lefevre wrote:
> > I would not associate a T-format with the interval type T.
> > For instance, for a binary64-based inf-sup interval type T, one may
> > want the midpoint in binary64, but also in binary128, as allowed by
> > IEEE 754 (§5.4.1).
>
> I proposed long ago that, ONLY for inf-sup types derived from
> IEEE754 formats, all _interval arithmetic operations_ should be
> available in "formatOf" mode (this is what §5.4.1 is about).
>
> Vincent, your comment suggests we should extend this feature from
> arithmetic operations to numeric functions of intervals. Would that
> do what you think is needed?
I think it would make sense and be consistent to interval arithmetic
operations and also the IEEE-754 formatOf operations.
> >>> ...
> >> (John Pryce wrote:)
> >>> Level 2 midpoint, etc., must return a datum of some
> >>> number format. Hence I see nothing for it but to make the revised
> >>> definition:
> >>>
> >>> An interval type T is a set of mathematical intervals,
> >>> plus a specified T-hull operation, plus a specified
> >>> number format, let's call it the T-format.
> >>>
> >>> For each implemented T, each numeric operation on intervals shall
> >>> have a T-version that returns a result of this T-format. (One might
> >>> allow different operations to return results of different formats,
> >>> but to me that seems way too complicated.)
> >>
> >> I would not associate a T-format with the interval type T.
>
> If we don't, how are we to define numeric functions of intervals,
> for types that are not inf-sup? For a general implicit type T there
> is NO intrinsic way to associate a number format; IMO it must be
> specified as part of T's definition.
The number format would have to be specified in the function. How it
is specified (possibly implicitly) would be implementation-defined
or language-defined. Basically, this means that we should not enforce
a privileged number format associated with a type T; a default number
format associated with a type T is allowed but should not be required.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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