Re: Revised version of Level 1 text (draft)
> Date: Wed, 11 May 2011 14:34:54 +0200
> From: Arnold Neumaier <Arnold.Neumaier@xxxxxxxxxxxx>
> To: Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>
> CC: Nate Hayes <nh@xxxxxxxxxxxxxxxxx>, stds-1788@xxxxxxxxxxxxxxxxx
> Subject: Re: Revised version of Level 1 text (draft)
>
> Dan Zuras Intervals wrote:
> > I am more interested in the 'natural' uses of set
> > theoretic functions. In the case of intersection,
> > they seem to involve deriving an interval for which the
> > properties of both operands are true. In the Newton's
> > iteration:
> >
> > given xx(k) & xmid (the midpoint of that
> > interval, we compute:
> >
> > xx(k+1) = (xmid - f(xmid)/ffprime(xx(k)))
> > \intersect xx(k)
> >
> > That is, we seek a new interval that is contained in
> > BOTH the old interval & the image of that interval
> > under Newton's map.
>
> This is an intersection of bare intervals. decorations make sense only
> for range computations, not for domain reductions, as here. Thus they
> are needed inside the computations of
> f(xmid) [xmid must be an interval to take care of roundoff]
> and
> ffprime(xx(k)),
> to check the assumptions under which the interval newton formula holds,
> but not for the other calculations in the above iteration.
I both disagree & agree.
I disagree in that the intersection in the Newton's
iteration involves both the domain & range of a fixed
point mapping.
But I agree in that the result must be used as a
domain in the next iteration &, as such, must be
wiped of any accumulated nastyness involved in the
fixed point calculation.
Some time ago, we discussed the need for some function
that wipes the decoration of an interval clean for just
this application.
In this sense the iteration might be more properly
written as something like:
xx(k+1) = makeSafe(xx(k) \intersect
(xmid - f(xmid)/ffprime(xx(k))))
where makeSafe() is the obvious decoration wiper.
Dan