Re: midpoint-radius representation
Svetoslav Markov schrieb:
On 11 Nov 2008 at 10:18, Arnold Neumaier wrote:
> Then let me state my arguments against including it as an intrinsic
representation of intervals in the standard.
(i) Specifying the knowledge x>=0 via mid-rad is impossible.
(ii) conversion from inf-sup to mid-rad and back imposes
a heavy loss of quality for wide intervals (as they are frequent
in applications to constrained global optimization).
these items refer to computer implementation and could be
better discussed by Siegfried.
(iii) Operations are awkward to define, even for addition.
In theory, midpoints and radii both add; however, the rounding error
incurred upon adding midpoints must be added to the radius.
The formulas for multiplication are messy and unduly expensive;
see Proposition 1.6.5 of my book ''Interval methods for systems
of equations''.
The above mentioned formulas are indeed not constructive.
They are constructive (obviously) and can be evaluated without any
case distinctions. The cost is
1 sign, 4 abs, 2 add, 6 mult, and inf and sup of 3 numbers each.
Simple formula is given in [...] the appended paper:
Markov S., D. Claudio: On the Interval Arithmetic in
Midpoint-Radius Form. Mathematics and Education
in Mathematics 33, 2004, Inst. Math. and Informatics, BAS, 434-439.
Note that the formula for multiplication (12) is extremely simple
Assuming exact arithmetic, I count for the product of two intervals
that happen not to contain zero (one of 4 cases) the evaluation of
2 sign, 4 abs, 2 add, 6 mult, 2 div,
of similar complexity as the formula from my Proposition 1.6.5,
but expensive compared to
<=4 mult and <= 1 max and 1 min of two numbers each
in the inf-sup representation. I wouldn't call this extremely simple.
Moreover, this does not yet address rounding errors. To cover these,
you need ordinary interval arithmetic to evaluate the intermediate
results, and get a problem with the case distinction in case that
one of the kappas evaluates to [1,1+eps].
Or how do you propose to account for rounding errors?
Arnold Neumaier