Re: the "set paradigm" is harmful
Arnold,
please see below my interpolated comments
to your remarks.
On 12 Feb 2009 at 13:17, Arnold Neumaier wrote:
> Svetoslav Markov schrieb:
> >
> > Surely, MR-presented intervals, resp. approximate numbers,
> > admit probabilistic interpretation, e.g. midpoint can
> > be mean value, in Gaussian distributions radius can be
> > standard deviation or half-width of confidence interval, etc.
> > (Note that the guaranteed containment property can be still there,
> > say within the 95% of probability).
>
> This is not true.
>
> if |x-x_0|<=sigma with probability 95% and
> |y-y_0|<=sigma' with probability 95%, there is no guarantee that
> the product x*y will lie in the product of the corresponding
> intervals with 95%.
>
I wrote "can be", not "is". What I say is true for widely used
probability operations such as addition and multiplication
by scalars.
>
> > Note that
> > MR-form intervals incorporate the set paradigm --- up to the
> > special cases (oo, a], [b, oo).
>
> Not if, as you suggest, the radius is represented by 2 digits only.
>
1-2-3 digit representation is suggested for NARROW intervals.
Midrad representation/computation is practically useful for narrow
intervals, the intervals that model approximate numbers.
> For simplicity, assume a decimal representation, and consider the
> interval xx=[0,202]. Its conversion to midrad form is ambiguous and
> overestimates. The three centered intervals <mid=9.2e1,rad=1.1e2>,
> <mid=1.01e2,rad=1.1e2>, and <mid=1.1e2,rad=1.1e2> are among the many
> minimal centered intervals with 2 digit radius containing xx,
> corresponding to the intervals [-18,202], [-9,211] and [0,220].
>
> One loses almost 9% of the accuracy (1.1/1.01=1.0891...), and all
> apart from the last version don't even preserve the sign information
> inherent in the infsup presentation.
>
> Also, the cost of a rigorous midrad interval arithmetic (including
> rounding error control) is twice that of infsup arithmetic.
> Proponents of midrad arithmetic should first remove this disatvantage.
>
Here it seems that you forget about the 2-digit mantissa
representation!
But I am glad that you admit that in real arithmetic the cost is same.
>
> > That is why MR-form is even more
> > important than sup-inf form.
>
> That is why the MR-form is useless for a standard.
> Anyone interested in using the midrad form can easily convert it
> at the beginning and end of a computation with the functions proposed
> in the Vienna Proposal. More is not needed.
>
IMO it is needed for fast computations with narrow intervals.
>
> > MR-form is more convenient than the
> > sup-inf form in many applications (possibly for Taylor polynomials
> > as well).
>
> If this is so, who is currently using the midrad form for executing
> interval operations in some of these many applications?
>
> Intlab provides an intermediate midrad representation
> in a fast matrix multiply option, but at the expense of an
> overestimation factor of 1.5 for the results widths.
> To get rid of this factor would make the midrad representation
> slower than the standard version!
>
again: narrow intervals & 2-digit mantissa!
>
> Some authors often use this form as an intrinsic form in
> > computations/proofs and then translate results in sup-inf form
> > (notably J. Rohn).
>
> Nobody says that mid and rad are not useful theoretical tools.
> But the standard is about standardizing implementations.
>
> Where is the fast and much-used rigorous implementation of
> midrad arithmetic that people use in their applications?
> I haven't heard of any.
>
who knows, may there will soon appear patents about midrad
computations ...
>
> > Conclusion is: MR-presentation of intervals deserves (at least) equal
> > consideration in the standard as the sup-inf form.
>
> This conclusion is completely unsupported by the facts.
facts? I cannot see them.
Regards,
Svetoslav