Re: the "set paradigm" is harmful
Svetoslav Markov schrieb:
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.
This allows not a single nonlinear computation with your
probability interpretation. So the practical impact will be minimal.
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.
But you had asserted ''up to the special cases (oo, a], [b, oo)''
Your new assertion amounts to ''upto the no longer special cases
where the interval is no longer narrow'' - which is something quite
different.
Or do you want to support a format where the number of digits for the
radius depends on the width of the interval?
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!
The metalevel computation of the 9% has nothing to do with such a
restriction on the interval computation.
But I am glad that you admit that in real arithmetic the cost is same.
No. As you can read, I was asserting (for optimal bounds)
a slowdown factor of 2 over infsup interval arithmetic for
multiplication.
(Rumps centered multiplication is not optimal.)
Do you want to have nonoptimal centered arithmetic, or optimal
noncentered arithmetic, or both?
Have you thought about how to compute optimal bounds for centered
arithmetic in the presence of rounding errors?
The literature does not seem to contain any discussion of this issue.
Arnold Neumaier