Revamping the numeric classes

[Ch. A. Herrmann]

moved to haskell-cafe

    Ketil> E.g. way back, I wrote a simple differential equation solver.
    Ketil> Now, the same function *could* have been applied to vector
    Ketil> functions, except that I'd have to decide on how to implement
    Ketil> all the "Num" stuff that really didn't fit well.  Ideally, a
    Ketil> nice class design would infer, or at least allow me to
    Ketil> specify, the mathematical constraints inherent in an
    Ketil> algorithm, and let my implementation work with any data
    Ketil> satisfying those constraints.

the problem is that the --majority, I suppose?-- of mathematicians
tend to overload operators. They use "*" for matrix-matrix
multiplication as well as for matrix-vector multiplication etc.

Therefore, a quick solution that implements groups, monoids, Abelian
groups, rings, Euclidean rings, fields, etc. will not be sufficient.

I don't think that it is acceptable for a language like Haskell
to permit the user to overload predefined operators, like "*".

A cheap solution could be to define a type MathObject and operators like 
   :*: MathObject -> MathObject -> MathObject
Then, the user can implement:

a :*: b = case (a,b) of
             (Matrix x, Matrix y) -> foo
             (Matrix x, Vector y) -> bar
-- 
 Christoph Herrmann
 E-mail:  herrmann@fmi.uni-passau.de
 WWW:     http://brahms.fmi.uni-passau.de/cl/staff/herrmann.html