Re: How do I bisect unbounded intervals?
Vincent et al,
On 01/17/2012 06:43 AM, Vincent Lefevre wrote:
On 2012-01-16 22:25:49 +0000, John Pryce wrote:
Dan, Vincent, Nate etc.
This discussion reminds me of a Level 2 issue that must be resolved.
When discussing finite precision it's clear you've all been assuming
an inf-sup representation. How do midpoint and the other numeric
functions of intervals work for an _implicit_ type?
Even for inf-sup, I now think that this is not clear: is the format
of the result necessarily the same as the number format associated
with the interval type?
It will be if we make it so. An advantage of that would be
clarity and simplicity. Also, since "mid" is a floating point
value, other standards could handle conversion of the target
type of the "mid." (just my own opinion)
I've basically defined an interval type T as a set of mathematical
intervals plus a specified T-hull operation. No number-format is
mentioned.
I'm a bit confused here. Vincent mentioned "number format associated
with the interval type".
But 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.
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).
That's an interesting thought. Can you elaborate on why one might
want more precision in mid than in the end points?
Baker