Re: Sorry, my example in error Re: Tetrits and "stickiness"
Baker, Dan, P1788
Further apologies for being behind...
On 15 Apr 2010, at 17:34, Ralph Baker Kearfott wrote:
> The Brouwer fixed point theorem is related to the contraction mapping theorem, but not exactly the
> same. (Does someone want to argue equivalence?)
I don't think they can be made equivalent. E.g., I think g(x,y) = (cos(100 y), cos(100 x)), on the square [-1,1] x [-1,1], satisfies the Brouwer conditions, but no way can it be a contraction map.
> The hypothesis of the theorem that is violated by the domainOut condition is continuity.
>
> 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.
Yes.
> 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.
Sounds sensible.
> 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.
No. I expressed a preference for a *single* flag meaning "both defined and continuous on the domain", but several people (I think including Juergen and Arnold) preferred two separate flags. I am still unclear why this is useful.
John