-- | Implement ordered pairs all over again for fun (and to make sure
--   that we can manipulate them algebraically). Also require (as
--   opposed to the built-in ordered pairs) that the elements have
--   matching types.
--
module TwoTuple
where

import Vector


data TwoTuple a = TwoTuple a a
  deriving (Eq)

instance (Show a) => Show (TwoTuple a) where
  show (TwoTuple x y) = "(" ++ (show x) ++ ", " ++ (show y) ++ ")"

instance Functor TwoTuple where
  f `fmap` (TwoTuple x1 y1) = TwoTuple (f x1) (f y1)

instance (RealFloat a) => Vector (TwoTuple a) where
  -- The standard Euclidean 2-norm. We need RealFloat for the square
  -- root.
  norm_2 (TwoTuple x y) = fromRational $ toRational (sqrt(x^2 + y^2))

  -- The infinity norm, i.e. the maximum entry.
  norm_infty (TwoTuple x y) =
    fromRational $ max absx absy
    where
      absx = abs (toRational x)
      absy = abs (toRational y)

-- | It's not correct to use Num here, but I really don't want to have
--   to define my own addition and subtraction.
instance Num a => Num (TwoTuple a) where
  -- Standard componentwise addition.
  (TwoTuple x1 y1) + (TwoTuple x2 y2) =
    TwoTuple (x1 + x2) (y1 + y2)

  -- Standard componentwise subtraction.
  (TwoTuple x1 y1) - (TwoTuple x2 y2) =
    TwoTuple (x1 - x2) (y1 - y2)

  -- Left undefined to prevent mistakes. One sane definition
  -- would be componentwise multiplication.
  (*) _ _ = error "multiplication of vectors is undefined"

  abs _ = error "absolute value of vectors is undefined"

  signum _ = error "signum of vectors is undefined"

  fromInteger x = TwoTuple (fromInteger x) (fromInteger x)