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.RealRing as RealRing
  17 import qualified Algebra.ToInteger as ToInteger
  18
  19 -- Since the norm is defined on a vector space, we should be able to
  20 -- add and subtract anything on which a norm is defined. Of course
  21 -- 'Num' is a bad choice here, but we really prefer to use the normal
  22 -- addition and subtraction operators.
  23 class Normed a where
  24   norm_p :: (ToInteger.C c, Algebraic.C b, Absolute.C b) => c -> a -> b
  25   norm_infty :: (RealField.C b) => a -> b
  26
  27   -- | The "usual" norm. Defaults to the Euclidean norm.
  28   norm :: (Algebraic.C b, Absolute.C b) => a -> b
  29   norm = norm_p (2 :: Integer)
  30
  31 -- Define instances for common numeric types.
  32 instance Normed Integer where
  33   norm_p _ = abs . fromInteger
  34   norm_infty = abs . fromInteger
  35
  36 instance Normed Rational where
  37   norm_p _ = abs . fromRational'
  38   norm_infty = abs . fromRational'
  39
  40 instance Epsilon e => Normed (BigFloat e) where
  41   norm_p _ = abs . fromRational' . toRational
  42   norm_infty = abs . fromRational' . toRational
  43
  44 instance Normed Float where
  45   norm_p _ = abs . fromRational' . toRational
  46   norm_infty = abs . fromRational' . toRational
  47
  48 instance Normed Double where
  49   norm_p _ = abs . fromRational' . toRational
  50   norm_infty = abs . fromRational' . toRational