{-# LANGUAGE FlexibleInstances #-}

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

import Data.Number.BigFloat

-- 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 (Num a) => Normed a where
  norm_p :: (Integral c, RealFrac b) => c -> a -> b
  norm_infty :: RealFrac b => a -> b

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

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

instance Normed Rational where
  norm_p _ = fromRational
  norm_infty = fromRational

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

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