{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE RebindableSyntax #-}

-- | The 'Normed' class represents elements of a normed vector
--   space. We define instances for all common numeric types.
module Normed
where

import BigFloat

import NumericPrelude hiding (abs)
import Algebra.Absolute (abs)
import qualified Algebra.Absolute as Absolute
import qualified Algebra.Algebraic as Algebraic
import qualified Algebra.RealField as RealField
import qualified Algebra.ToInteger as ToInteger

-- Since the norm is defined on a vector space, we should be able to
-- add and subtract anything on which a norm is defined. Of course
-- 'Num' is a bad choice here, but we really prefer to use the normal
-- addition and subtraction operators.
class Normed a where
  norm_p :: (ToInteger.C c, Algebraic.C b, Absolute.C b) => c -> a -> b
  norm_infty :: (RealField.C b) => a -> b

  -- | The "usual" norm. Defaults to the Euclidean norm.
  norm :: (Algebraic.C b, Absolute.C b) => a -> b
  norm = norm_p (2 :: Integer)

-- Define instances for common numeric types.
instance Normed Integer where
  norm_p _ = abs . fromInteger
  norm_infty = abs . fromInteger

instance Normed Rational where
  norm_p _ = abs . fromRational'
  norm_infty = abs . fromRational'

instance Epsilon e => Normed (BigFloat e) where
  norm_p _ = abs . fromRational' . toRational
  norm_infty = abs . fromRational' . toRational

instance Normed Float where
  norm_p _ = abs . fromRational' . toRational
  norm_infty = abs . fromRational' . toRational

instance Normed Double where
  norm_p _ = abs . fromRational' . toRational
  norm_infty = abs . fromRational' . toRational