Re: mid-rad, inf-sup, a caution...
Svetoslav Markov wrote:
On 11 May 2010 at 9:19, John Pryce wrote:
Svetoslav, you write
Moreover, many facts lead me to conclude that mid-rad is a "primary type"
deserving a "first class treatment"!
Well, will you please give some of these facts?
I have in mind some facts like:
- the simplicity of computations that engineers usually perform
by hand using the main value (midpoint) and a few digists
for the error bound. See. e. g. Sunaga paper.
They do not perform the standard midrad arithmeitc, since they always
ignore higher order terms. Eg., they calculate
if y=f(x) then f(x+-r)=y+-f'(x))r,
which is not usually a valid enclosure. All rigor is lost.
How would you do the case of f(x)=sqrt(x) rigorously with midrad?
- the few digits are really very few, say 1-3, see e. g. :
van Emden, M. H., On the Significance of Digits in
Interval Notation, Reliable Computing 10, 1, 45--58.
This does not translate into speed on current hardware, but may
costs a lot of accuracy (compared to standard infsup) when evaluating
deeply nested expressions such as continued fractions.
- the simplicity of the mid-rad formulae for arithmetic operations
vs sup-inf formulae;
This is only in exact arithmetic. If roundoff must be correctly
addressed, the formulas get messy!
Standard functions also are messy:
The only simple way to implement them is to convert to infsup,
execuute the oeration, and then convert back.
Is there any existing rigorous implementation of midrad arithmetic
for the usual standard functions? If not (which I suspect is the case),
this already tells the full story!
- Who will benefit?
IMO everybody will benefit. It is simpler to learn and use
mid-rad arithmetic, than inf-sup one.
No. There are no simple formulas for the result of standard functions.
Even multiplication is simple to learn only when ignoring roundoff.
Arnold Neumaier