src/Normed.hs

   1 {-# LANGUAGE FlexibleInstances #-}
   2
   3 -- | The 'Normed' class represents elements of a normed vector
   4 --   space. We define instances for all common numeric types.
   5 module Normed
   6 where
   7
   8 import Data.Number.BigFloat
   9
  10 -- Since the norm is defined on a vector space, we should be able to
  11 -- add and subtract anything on which a norm is defined. Of course
  12 -- 'Num' is a bad choice here, but we really prefer to use the normal
  13 -- addition and subtraction operators.
  14 class (Num a) => Normed a where
  15   norm_p :: (Integral c, RealFrac b) => c -> a -> b
  16   norm_infty :: RealFrac b => a -> b
  17
  18   -- | The "usual" norm. Defaults to the Euclidean norm.
  19   norm :: RealFrac b => a -> b
  20   norm = norm_p (2 :: Integer)
  21
  22 -- Define instances for common numeric types.
  23 instance Normed Integer where
  24   norm_p _ = fromInteger
  25   norm_infty = fromInteger
  26
  27 instance Normed Rational where
  28   norm_p _ = fromRational
  29   norm_infty = fromRational
  30
  31 instance Epsilon e => Normed (BigFloat e) where
  32   norm_p _ = fromRational . toRational
  33   norm_infty = fromRational . toRational
  34
  35 instance Normed Double where
  36   norm_p _ = fromRational . toRational
  37   norm_infty = fromRational . toRational