[NumericPrelude] Progress toward UFD implementation
Henning Thielemann
lemming at henning-thielemann.de
Wed Jun 30 07:08:32 EDT 2010
On Tue, 29 Jun 2010, Lewis-Sandy, Darrell wrote:
> I thought I should take a moment and update you on my progress toward implementing the proposed new classes for the numeric prelude. The three classes which I proposed to disambiguate the PID class are those below:
>
> {- from Algebra.UniqueFactorizationDomain.hs -}
> class (Units.C a, ZeroTestable.C a, Integral.C a) => C a where
> factors :: a -> [a] -- factor a value into its unique factorization
> isReducible :: a -> Bool
> gcdCofacts :: [a] -> (a,[a])
> gcd :: [a] -> a
> cofacts :: [a] -> [a]
> lcm :: [a] -> a
>
> {- from PrincipalIdealDomain.hs -}
> class (UFD.C a, Units.C a, ZeroTestable.C a) => C a where
> basis :: [a] -> a -- the basis of an ideal
Is the basis always a single element?
> {- from Algebra.EuclideanDomain.hs -}
> class (Units.C a, PID.C a, ZeroTestable.C a) => C a where
> valuation :: a -> Integer -- a Euclid valuation
> extendedEuclid :: a -> a -> (a,(a,a))
> euclid :: a -> a -> a
>
> Have been implemented, and the instancing for a large number of numeric
> types (atomic types, Complex, Gaussian integers, Complex numbers) have
> been completed. I have begun work on implementing the instances for
> polynomials (over integer types), and have just finished debugging the
> Hensel lifting routine.
Great! Would you provide automated tests for the non-trivial algorithms?
> Unlike the beautiful constructive proofs, the code is lacking in
> elegance (due largely to my hacks for modular arithmetic).
Sure, an elegant implementation would be nicer.
> I thought that this could be improved by implementing Zp as a data type
> with appropriate class instances. However, it seems like there is a
> conservation of difficulty principle here -- all the different
> approaches which I have investigated to date (up to and including
> dependent types) appear to have drawbacks. I noticed that you had
> implemented both residue classes and GF(2^32-5) and was hoping that you
> might have some thoughts on this subject.
Yes, there is no perfect solution, thus I decided to implement the
basic functionality without fitting into any numeric type class as in
http://hackage.haskell.org/packages/archive/numeric-prelude/0.1.3.4/doc/html/Number-ResidueClass.html
and then I provide different interfaces to those functions in modules in
a sub-directory. Maybe I should split also modules like Polynomial this
way, since the basic polynomial operations are also needed somewhere else,
without the need to refer to polynomials explicitly.
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