```{-# LANGUAGE CPP #-}
{-# LANGUAGE DefaultSignatures #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  Linear.Vector
-- Copyright   :  (C) 2012 Edward Kmett
-- Maintainer  :  Edward Kmett <ekmett@gmail.com>
-- Stability   :  provisional
-- Portability :  portable
--
-- Operations on free vector spaces.
-----------------------------------------------------------------------------
module Linear.Vector
, negated
, (^*)
, (*^)
, (^/)
, basis
, basisFor
) where

import Control.Applicative
import Data.Complex
import Data.Foldable (foldMap)
import Data.Functor.Bind
import Data.HashMap.Lazy as HashMap
import Data.Hashable
import Data.IntMap as IntMap
import Data.Map as Map
import Data.Monoid (Sum(..))
import Data.Traversable (Traversable, mapAccumL)
import Linear.Instances ()

-- \$setup
-- >>> import Control.Lens
-- >>> import Linear.V2

infixl 6 ^+^, ^-^
infixl 7 ^*, *^, ^/

class Bind f => Additive f where
-- | The zero vector
zero :: Num a => f a
#ifndef HLINT
default zero :: (Applicative f, Num a) => f a
zero = pure 0
#endif

-- | Compute the sum of two vectors
--
-- >>> V2 1 2 ^+^ V2 3 4
-- V2 4 6
(^+^) :: Num a => f a -> f a -> f a
#ifndef HLINT
default (^+^) :: (Applicative f, Num a) => f a -> f a -> f a
(^+^) = liftA2 (+)
{-# INLINE (^+^) #-}
#endif

-- | Compute the difference between two vectors
--
-- >>> V2 4 5 - V2 3 1
-- V2 1 4
(^-^) :: Num a => f a -> f a -> f a
#ifndef HLINT
default (^-^) :: (Applicative f, Num a) => f a -> f a -> f a
(^-^) = liftA2 (-)
{-# INLINE (^-^) #-}
#endif

-- | Linearly interpolate between two vectors.
lerp :: Num a => a -> f a -> f a -> f a
lerp alpha u v = alpha *^ u ^+^ (1 - alpha) *^ v
{-# INLINE lerp #-}

zero = IntMap.empty
(^+^) = IntMap.unionWith (+)
xs ^-^ ys = IntMap.unionWith (+) xs (negated ys)

instance Ord k => Additive (Map k) where
zero = Map.empty
(^+^) = Map.unionWith (+)
xs ^-^ ys = Map.unionWith (+) xs (negated ys)

instance (Eq k, Hashable k) => Additive (HashMap k) where
zero = HashMap.empty
(^+^) = HashMap.unionWith (+)
xs ^-^ ys = HashMap.unionWith (+) xs (negated ys)

-- | Compute the negation of a vector
--
-- >>> negated (V2 2 4)
-- V2 (-2) (-4)
negated :: (Functor f, Num a) => f a -> f a
negated = fmap negate
{-# INLINE negated #-}

-- | Compute the left scalar product
--
-- >>> 2 *^ V2 3 4
-- V2 6 8
(*^) :: (Functor f, Num a) => a -> f a -> f a
(*^) a = fmap (a*)
{-# INLINE (*^) #-}

-- | Compute the right scalar product
--
-- >>> V2 3 4 ^* 2
-- V2 6 8
(^*) :: (Functor f, Num a) => f a -> a -> f a
f ^* a = fmap (*a) f
{-# INLINE (^*) #-}

-- | Compute division by a scalar on the right.
(^/) :: (Functor f, Fractional a) => f a -> a -> f a
f ^/ a = fmap (/a) f
{-# INLINE (^/) #-}

-- @setElement i x v@ sets the @i@'th element of @v@ to @x@.
setElement :: Traversable t => Int -> a -> t a -> t a
setElement i x = snd . mapAccumL aux 0
where aux j y = let j' = j + 1
y' = if i == j then x else y
in j' `seq` (j', y')

-- | Produce a default basis for a vector space. If the dimensionality
-- of the vector space is not statically known, see 'basisFor'.
basis :: (Applicative t, Traversable t, Num a) => [t a]
basis = [ setElement k 1 z | k <- [0..n - 1] ]
where z = pure 0
n = getSum \$ foldMap (const (Sum 1)) z

-- | Produce a default basis for a vector space from which the
-- argument is drawn.
basisFor :: (Traversable t, Enum a, Num a) => t a -> [t a]
basisFor v = [ setElement k 1 z | k <- [0..n-1] ]
where z = 0 <\$ v
n = getSum \$ foldMap (const (Sum 1)) v
```