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.