Re: Multi-precision (was...Please give me advice)
Vincent Lefevre wrote:
On 2010-10-15 12:44:15 +0200, Arnold Neumaier wrote:
Vincent Lefevre wrote:
On 2010-10-14 17:43:19 +0200, Arnold Neumaier wrote:
For me, the criterion distinguishing between good and bad
representations is whether they can represent highly asymmetric
intervals such as [1,1e6] or [-1,inf] without much overestimation.
Why? If one uses valid mode, an implementation could still return the
same result for f([-1,inf]) and f(Entire). So, requiring [-1,inf] to
be representable exactly would be useless.
Of course, one can implement all operations to give the result Entire.
Then requiring any interval besides Entire to be representable exactly
would be useless.
This is not what I've said.
I know, but your argument applies to this caricature as well.
Being useless in some circumstances is no excuse for losing
information which is important in others.
Only unbounded intervals (and maybe some
other large intervals) would be seen as Entire, and this wouldn't be
a problem for applications that use IA to certify a result that could
be computed in FP arithmetic otherwise.
But I assumed that a reasonable standard would impose at least some
sensible requirements on accuracy.
Well, [-1,inf] is far from being an accurate representation of a real
number!
It is an accurate representation of a connected, real domain.
The accuracy requirement is one of not unnecessarily widening
the range of a result that depends on wide arguments.
Intervals often do not represent an inaccurate number but
(and indeed often) a range of possible numbers.
For example, in robotics, an interval represents all possible
numbers that a given control is allowed to take. One needs to
verify bounds on certain critical functions that depend on these
controls in order to guarantee their safe working.
In this example, one has of course no infinite bounds.
But the latter occur naturally in many other applications,
beginning with linear programs (which arise virtually everywhere).
Therefore good interval software must have a good behavior on
wide intervals, too.
In particular, one can and should require that all arithmetic
operations +-*/ preserve the sign of the optimal result.
This is something very arbitrary. For instance, a mid-rad implementation
(that doesn't have this property) could give a better accurary than
an inf-sup implementation that uses a format of the same size (because
a mid-rad implementation could use a higher precision for mid). Thus
I disagree.
A standard must ensure a minimum quality for all interesting
applications, not only for those where an interval just measures
the inaccuracy in a not exactly computed number.
Arnmold Neumaier