Re: mid-rad, inf-sup, a caution...
Svetoslav Markov wrote:
I realise that few people are familiar with
mid-rad computations. Sunaga original
paper is probably the best source.
This shows how little midrad has been used and developed in
the over 50 yars since Sunaga's work. it has always been a
minority enterprise within the interval community.
Midrad has always been only a research subject, not a tool
for efficiency in interval applications.
For engineers it will be interesting to see how
Sunaga computes rigorously in mid-rad arithmetic.
We reworded his method in our paper:
Markov, S., Okumura, K.: The Contribution of T. Sunaga to
Interval Analysis and Reliable Computing, in: Csendes, T. (ed.),
Developments in Reliable Computing, Kluwer, 1999, 167--188.
There in section:
3. Sunaga’s Proposal for Application of Interval Analysis to Reliable
Numerical Computation
we described Sunaga's rigorous method. I strogly recommend
engineering specialist to look at Sunaga's method. I attach
a pre-published copy of the paper (without photo of Sunaga).
They will be very disappointed. Only rational functions are treated.
But engineers need square roots, exponentials, logarithms, sines, etc..
Even for rational functions, the case made is poor when compared to
infsup arithmetic:
Theorem 13 treats rounding, and formula (4) afterwards exact
multiplication of midrad intervals with nonnegative factors.
Combining both, one sees that to get the product of two nonnegative
intervals, one needs to compute with directed rounding 5 products
and 4 additions. 4+2 are directly visible, but the midpoint must be
computed twice to get upper and lower bounds, then one of them must
be chosen as the actual midpoint, the difference must be computed
and then added to the radius. (With some extra trickery, one
multiplication and addition can be saved.)
Where is the claimed simplicity compared to the infsup formula
[a,b]*[c,d]=[ac,bd]
with the first product rounded down, the second rounded up?
Trying to do the interval square root shows that this is not just
a poor feature of multiplication....
Arnold Neumaier