src/Normed.hs

   1 {-# LANGUAGE FlexibleInstances #-}
   2 {-# LANGUAGE RebindableSyntax #-}
   3
   4 -- | The 'Normed' class represents elements of a normed vector
   5 --   space. We define instances for all common numeric types.
   6 module Normed
   7 where
   8
   9 import BigFloat
  10
  11 import NumericPrelude hiding (abs)
  12 import Algebra.Absolute (abs)
  13 import qualified Algebra.Absolute as Absolute
  14 import qualified Algebra.Algebraic as Algebraic
  15 import qualified Algebra.RealField as RealField
  16 import qualified Algebra.ToInteger as ToInteger
  17
  18 -- Since the norm is defined on a vector space, we should be able to
  19 -- add and subtract anything on which a norm is defined. Of course
  20 -- 'Num' is a bad choice here, but we really prefer to use the normal
  21 -- addition and subtraction operators.
  22 class Normed a where
  23   norm_p :: (ToInteger.C c, Algebraic.C b, Absolute.C b) => c -> a -> b
  24   norm_infty :: (RealField.C b) => a -> b
  25
  26   -- | The "usual" norm. Defaults to the Euclidean norm.
  27   norm :: (Algebraic.C b, Absolute.C b) => a -> b
  28   norm = norm_p (2 :: Integer)
  29
  30 -- Define instances for common numeric types.
  31 instance Normed Integer where
  32   norm_p _ = abs . fromInteger
  33   norm_infty = abs . fromInteger
  34
  35 instance Normed Rational where
  36   norm_p _ = abs . fromRational'
  37   norm_infty = abs . fromRational'
  38
  39 instance Epsilon e => Normed (BigFloat e) where
  40   norm_p _ = abs . fromRational' . toRational
  41   norm_infty = abs . fromRational' . toRational
  42
  43 instance Normed Float where
  44   norm_p _ = abs . fromRational' . toRational
  45   norm_infty = abs . fromRational' . toRational
  46
  47 instance Normed Double where
  48   norm_p _ = abs . fromRational' . toRational
  49   norm_infty = abs . fromRational' . toRational