Re: How do I bisect unbounded intervals?
> Subject: Re: How do I bisect unbounded intervals?
> From: John Pryce <j.d.pryce@xxxxxxxxxxxx>
> Date: Mon, 16 Jan 2012 22:25:49 +0000
> To: stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
>
> 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?
Nate & I have had a lively offline discussion over the weekend
on these issues in so far as they relate to the uses of these
functions in various interval methods. Needless to say, Nate
has been the B&B guy & I bring some floating-point experience
to the table.
To answer your question as asked, implicit forms of midpoint()
& width() are trivial to define but difficult to use in that
they take place in an arithmetic that has no simple (useful)
notion of a ULP. Thus it is difficult for B&Bs (&, I suspect,
other methods as well) to define simple, robust, & WORKING
termination conditions that cover the space of possible
problems from entire & the semi-infinites all the way down to
the narrowest of intervals.
And yet, I believe our discussion has borne fruit & Nate & I
have fairly simple definitions of width(), midpoint(), & ULP()
for inf-sup intervals for which Nate has a set of terminations
conditions that seem robust enough to work everywhere.
I will ask for Nate's forgiveness at this point as I am kinda
"outing" our discussion which is not done yet. But we just had
a breakthrough & I think we will have something to propose soon.
The disappointing thing is that what seems easy to do among the
inf-sups seems difficult or impossible to do among the mid-rads
& those intervals with variable precision. Why will become
clear when we go public but I think better minds than I will be
needed for the implicits.
>
> 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. 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.)
>
> Your views please. Any further complications (Ugh!) that need to be made?
>
> I'll comment on the midpoint/bisect discussion separately.
>
> John Pryce
I'm not sure how this relates to the issue. But take a
look at our work before you go too far. I think it solves
many of the issues here.
How's THAT for a teaser. :-)
Dan