src/Normed.hs

   1 {-# LANGUAGE FlexibleInstances #-}
   2 {-# LANGUAGE RebindableSyntax #-}
   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 BigFloat
   9
  10 import NumericPrelude hiding (abs)
  11 import Algebra.Absolute
  12 import Algebra.Field
  13 import Algebra.Ring
  14 import Algebra.ToInteger
  15
  16 -- Since the norm is defined on a vector space, we should be able to
  17 -- add and subtract anything on which a norm is defined. Of course
  18 -- 'Num' is a bad choice here, but we really prefer to use the normal
  19 -- addition and subtraction operators.
  20 class (Algebra.Ring.C a, Algebra.Absolute.C a) => Normed a where
  21   norm_p :: (Algebra.ToInteger.C c,
  22              Algebra.Field.C b,
  23              Algebra.Absolute.C b)
  24             => c -> a -> b
  25
  26   norm_infty :: (Algebra.Field.C b,
  27                  Algebra.Absolute.C b)
  28                 => a -> b
  29
  30   -- | The "usual" norm. Defaults to the Euclidean norm.
  31   norm :: (Algebra.Field.C b, Algebra.Absolute.C b) => a -> b
  32   norm = norm_p (2 :: Integer)
  33
  34 -- Define instances for common numeric types.
  35 instance Normed Integer where
  36   norm_p _ = abs . fromInteger
  37   norm_infty = abs . fromInteger
  38
  39 instance Normed Rational where
  40   norm_p _ = abs . fromRational'
  41   norm_infty = abs . fromRational'
  42
  43 instance Epsilon e => Normed (BigFloat e) where
  44   norm_p _ = abs . fromRational' . toRational
  45   norm_infty = abs . fromRational' . toRational
  46
  47 instance Normed Float where
  48   norm_p _ = abs . fromRational' . toRational
  49   norm_infty = abs . fromRational' . toRational
  50
  51 instance Normed Double where
  52   norm_p _ = abs . fromRational' . toRational
  53   norm_infty = abs . fromRational' . toRational