Comments on: The Role of the Interval Domain in Modern Exact Real Arithmetic http://math.andrej.com/2007/09/18/the-role-of-the-interval-domain-in-modern-exact-real-airthmetic/ Mathematics for computers Mon, 30 Jan 2012 07:14:48 +0000 hourly 1 http://wordpress.org/?v=3.2-beta2-18055 By: Andrej Bauer http://math.andrej.com/2007/09/18/the-role-of-the-interval-domain-in-modern-exact-real-airthmetic/comment-page-1/#comment-13891 Andrej Bauer Wed, 29 Sep 2010 22:06:50 +0000 http://math.andrej.com/2007/09/18/the-role-of-the-interval-domain-in-modern-exact-real-airthmetic/#comment-13891 @Edward: I have not looked at Taylor models. I know about problems with MPFR and GMP allocator. Has anyone got an idea how we could fix that? @Edward: I have not looked at Taylor models. I know about problems with MPFR and GMP allocator. Has anyone got an idea how we could fix that?

]]> By: Edward Kmett http://math.andrej.com/2007/09/18/the-role-of-the-interval-domain-in-modern-exact-real-airthmetic/comment-page-1/#comment-13886 Edward Kmett Wed, 29 Sep 2010 21:17:57 +0000 http://math.andrej.com/2007/09/18/the-role-of-the-interval-domain-in-modern-exact-real-airthmetic/#comment-13886 In addition to interval arithmetic, have you have a chance to look at Taylor models? These basically consist of the first n terms of a Taylor series and an interval to contain the slop and you can push the rounding error on the higher order coefficients into the interval. Berz and Makino have done a lot of work in this space. The advantage is that while the intervals expand, they are vastly smaller in this arrangement, and there exist tricks for reconditioning the Taylor model. I started to build a library in Haskell for working with these, but it used MPFR, which depended on GMP, and the Haskell MPFR bindings are broken due to internal changes/bug fixes in MPFR making it lean more heavily on the GMP allocator in a way that makes it incompatible with GHC's rewiring of that very same allocator. In addition to interval arithmetic, have you have a chance to look at Taylor models?

These basically consist of the first n terms of a Taylor series and an interval to contain the slop and you can push the rounding error on the higher order coefficients into the interval. Berz and Makino have done a lot of work in this space.

The advantage is that while the intervals expand, they are vastly smaller in this arrangement, and there exist tricks for reconditioning the Taylor model.

I started to build a library in Haskell for working with these, but it used MPFR, which depended on GMP, and the Haskell MPFR bindings are broken due to internal changes/bug fixes in MPFR making it lean more heavily on the GMP allocator in a way that makes it incompatible with GHC’s rewiring of that very same allocator.

]]> By: Andrej Bauer http://math.andrej.com/2007/09/18/the-role-of-the-interval-domain-in-modern-exact-real-airthmetic/comment-page-1/#comment-13881 Andrej Bauer Wed, 29 Sep 2010 19:24:53 +0000 http://math.andrej.com/2007/09/18/the-role-of-the-interval-domain-in-modern-exact-real-airthmetic/#comment-13881 Thanks for the link. Let me just point out that what we are talking about here builds on top of the usual interval arithmetic. That is to say, an implementation of real numbers, as envisaged in the paper, is a kind of self-adapting interval arithmetic with arbitrary-precision floats. Thanks for the link. Let me just point out that what we are talking about here builds on top of the usual interval arithmetic. That is to say, an implementation of real numbers, as envisaged in the paper, is a kind of self-adapting interval arithmetic with arbitrary-precision floats.

]]> By: anonymous http://math.andrej.com/2007/09/18/the-role-of-the-interval-domain-in-modern-exact-real-airthmetic/comment-page-1/#comment-13875 anonymous Wed, 29 Sep 2010 14:13:49 +0000 http://math.andrej.com/2007/09/18/the-role-of-the-interval-domain-in-modern-exact-real-airthmetic/#comment-13875 Look into: "Interval Constraint Plotting for Interactive Visual Exploration of Implicitly Defined Relations" T. Hickey, Z. Qiu, and M.H. van Emden Reliable Computing Vol. 6, No. 1, 2000, as part of a special issue on Reliable Geometric Computations, <a href="http://interval.sourceforge.net/interval/index.html" rel="nofollow">http://interval.sourceforge.net/interval/index.html</a>. Van Emden did a bunch of interval constraint stuff on floats. Basically interval arithmetic explodes into large intervals too quickly so they used constraint solvers to turn the answers into sets of intervals. Look into:

“Interval Constraint Plotting for Interactive Visual Exploration of Implicitly Defined Relations”
T. Hickey, Z. Qiu, and M.H. van Emden Reliable Computing Vol. 6, No. 1, 2000, as part of a special issue on Reliable Geometric Computations, http://interval.sourceforge.net/interval/index.html. Van Emden did a bunch of interval constraint stuff on floats. Basically interval arithmetic explodes into large intervals too quickly so they used constraint solvers to turn the answers into sets of intervals.

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