In hoc signo vinces (Was: Revamping the numeric classes)

[Joe Fasel]

On 09-Feb-2001 William Lee Irwin III wrote:
| Matrix rings actually manage to expose the inappropriateness of signum
| and abs' definitions and relationships to Num very well:
| 
| class  (Eq a, Show a) => Num a  where
|     (+), (-), (*)   :: a -> a -> a
|     negate          :: a -> a
|     abs, signum     :: a -> a
|     fromInteger     :: Integer -> a
|     fromInt         :: Int -> a -- partain: Glasgow extension
| 
| Pure arithmetic ((+), (-), (*), negate) works just fine.
| 
| But there are no good injections to use for fromInteger or fromInt,
| the type of abs is wrong if it's going to be a norm, and it's not
| clear that signum makes much sense.

For fromInteger, fromInt, and abs, the result should be a scalar matrix.
For the two coercions, I don't think there would be much controversy about this.
I agree that it would be nice if abs could return a scalar, but this requires
multiparameter classes, so we have to make do with a scalar matrix.

We already have this problem with complex numbers:  It might be nice
if the result of abs were real.

signum does make sense.  You want abs and signum to obey these laws:

        x == abs x * signum x
        abs (signum x) == (if abs x == 0 then 0 else 1)

Thus, having fixed an appropriate matrix norm, signum is a normalization
function, just as with reals and complexes.

If we make the leap to multiparameter classes, I think this is
the signature we want:

        class (Eq a, Show a) => Num a b | a --> b where
            (+), (-), (*)       :: a -> a -> a
            negate              :: a -> a
            abs                 :: a -> b
            signum              :: a -> a
            scale               :: b -> a -> a
            fromInteger         :: Integer -> a
            fromInt             :: Int -> a

Here, b is the type of norms of a.  Instead of the first law above, we have

        x == scale (abs x) (signum x)

All this, of course, is independent of whether we want a more proper
algebraic class hierarchy, with (+) introduced by Monoid, negate and (-)
by Group, etc.

Cheers,
--Joe

Joseph H. Fasel, Ph.D.              email: jhf@lanl.gov
Technology Modeling and Analysis    phone: +1 505 667 7158
University of California            fax:   +1 505 667 2960
Los Alamos National Laboratory      post:  TSA-7 MS F609; Los Alamos, NM 87545