Mitar wrote:
Hi!
First, disclaimer: everything I know about interval arithmetics comes
from this video:
http://video.google.com/videoplay?docid=-2285617608766742834
The discussion in the implementation of the Boost Interval Arithmetic
Library is also useful.
http://www.boost.org/libs/numeric/interval/doc/interval.htm
I would like to know if there is any implementation of interval
arithmetics in Haskell? I would like to play a little with that. I
checked the web and the straightforward approach I found:
http://cs.guc.edu.eg/faculty/sabdennadher/Publikationen/paper-wflp99.ps.gz
has from my point of view invalid implementation. For example, lower
bound in the sum should not be just calculated as the sum of lower
bounds of summands. It should return the greatest representable number
which is smaller or equal to the exact value of the sum. With just
boldly making a sum we ignore any errors we could introduce and this
is somehow against the idea of interval arithmetics.
Correct.
And as it is said at the end of the talk, a system behind interval
arithmetics should do a lot of work to make those intervals as small
as possible while still correct and counting in all the errors we
accumulated.
I think a strict-typed and lazy language like Haskell would be a good
place to implement this. But I would like to know if this would be
possible to do from the language itself, without making changes to the
compiler and/or runtime itself? Because, for example, a good
implementation should reformulate equations at the runtime accordingly
to exact values it wants to compute them on.
For intervals of floating point numbers you will need to gain access to
the FPU rounding modes for the machine (see some discussion here
http://www.boost.org/libs/numeric/interval/doc/rounding.htm). I don't
think that that is provided in the basic libraries so you would need to
use the FFI to get it from C. And proper rounding isn't available on all
platforms.
For unbounded values defined in Haskell (like Integer and Rational) you
need to provide round-down and round-up versions of operations that
won't produce exact answers (like division on Integer and sqrt on
Rational). Probably some sort of clever type-class setup could be used
to provide rounding functions for intervals (the same way Ix does clever
indexing for Array).
Has it been done already?
Sorry, not that I know of.
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