Re: Sorry, my example in error Re: Tetrits and "stickiness"
> Date: Thu, 15 Apr 2010 11:34:43 -0500
> From: Ralph Baker Kearfott <rbk@xxxxxxxxxxxx>
> To: Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>
> CC: stds-1788@xxxxxxxxxxxxxxxxx
> Subject: Re: Sorry, my example in error Re: Tetrits and "stickiness"
>
> Dan,
>
> The Brouwer fixed point theorem is related to the contraction
> mapping theorem, but not exactly the same. (Does someone want
> to argue equivalence?) The hypothesis of the theorem that is
> violated by the domainOut condition is continuity.
I understand.
>
> Although one sometimes does iterate a process, and fixed point
> iteration on g would indeed catch the problem, one also sometimes
> uses such computational fixed point theory to simply prove existence
> and uniqueness within a large box, without iterating to convergence.
> (For example, we sometimes construct as large a box as possible
> in which the conditions hold, to be able to eliminate it from
> further search in a branch and bound process.) Thus, checking the
> domainOut condition after the final result giving the overall
> expression value would be necessary, and we cannot rely on
> subsequent iterations revealing a problem.
>
> I don't view the DomainOut condition as a serious programming
> blunder in this context, but halfway between an operation exception
> and something like the "inexact" flag from 754. Tracing where it
> occurs could be useful, and the user could check its value after
> each operation, but checking a final sticky value might be more
> efficient.
Hmm. I would have picked the 'invalid' flag as the 754
analog in this case but no matter.
>
> By the way, for these fixed point questions, John had initially
> proposed a "continuous" flag. I think (and John, am I remembering
> correctly?) he decided these domain checks would fill the bill.
>
> Baker
>
If continuity is violated by the domainOut condition,
is it not sufficient to branch & bound until you either
eliminate all sub intervals or find one that does not
violate the domain decoration? In the former case, we
have no fixed point & in the latter, we have localized
it.
On the notion of a 'continuous' decoration: I believe
we have to be careful with its definition. While it is
possible that the underlying function on the Reals can
be flagged continuous or discontinuous as the case may
be, we are really defining functions on a finite set of
points (or intervals) that have a lower bound on just
how dense they can be (according to our working
precision).
Thus we can have examples of both continuous &
discontinuous functions that appear to be the opposite
when realized in floating-point.
For example, the sine function is perfectly continuous
over all of the Reals but as soon as the interval values
exceed a few times b^p the relationship between the sines
of consecutive representable numbers can take on a random
& discontinuous appearence. Then, it can falsely appear
continuous again for some b^q for q > p if b^q is close
to some large multiple of 2pi. Indeed, from something
like 2*pi*b^p onward, the sine of any non-point interval
will be [-1,1] & the sine of any point interval has pretty
much no meaning WRT a convergence method of any kind.
So while we may define 'continuous' or even NEED to
define it, we may not recognise it when we see it unless
the definition is very carefully crafted.
Just a warning, really. I think we will need such a
decoration in the end.
Just another opinion freely given & worth all you paid
for it. :-)
Dan