{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeFamilies #-}

module Linear.Vector
where

import Data.List (intercalate)
import Data.Vector.Fixed (
  Dim,
  Fun(..),
  N1,
  N2,
  N3,
  N4,
  Vector(..),
  construct,
  inspect,
  toList,
  )
import qualified Data.Vector.Fixed as V (
  length,
  )

import Normed


-- * Low-dimension vector wrappers.
--
-- These wrappers are instances of 'Vector', so they inherit all of
-- the userful instances defined above. But, they use fixed
-- constructors, so you can pattern match out the individual
-- components.

data D1 a = D1 a
type instance Dim D1 = N1
instance Vector D1 a where
  inspect (D1 x) (Fun f) = f x
  construct = Fun D1

data D2 a = D2 a a
type instance Dim D2 = N2
instance Vector D2 a where
  inspect (D2 x y) (Fun f) = f x y
  construct = Fun D2

data D3 a = D3 a a a
type instance Dim D3 = N3
instance Vector D3 a where
  inspect (D3 x y z) (Fun f) = f x y z
  construct = Fun D3

data D4 a = D4 a a a a
type instance Dim D4 = N4
instance Vector D4 a where
  inspect (D4 w x y z) (Fun f) = f w x y z
  construct = Fun D4


-- | Unsafe indexing.
--
--   Examples:
--
--   >>> let v1 = Vec2D 1 2
--   >>> v1 ! 1
--   2
--
(!) :: (Vector v a) => v a -> Int -> a
(!) v1 idx = (toList v1) !! idx

-- | Safe indexing.
--
--   Examples:
--
--   >>> let v1 = Vec3D 1 2 3
--   >>> v1 !? 2
--   Just 3
--   >>> v1 !? 3
--   Nothing
--
(!?) :: (Vector v a) => v a -> Int -> Maybe a
(!?) v1 idx
  | idx < 0 || idx >= V.length v1 = Nothing
  | otherwise                     = Just $ v1 ! idx




--instance (RealFloat a, Ord a, Vector v a) => Normed (Vn v a) where
  -- | The infinity norm. We don't use V.maximum here because it
  --   relies on a type constraint that the vector be non-empty and I
  --   don't know how to pattern match it away.
  --
  --   Examples:
  --
  --   >>> let v1 = make3d (1,5,2)
  --   >>> norm_infty v1
  --   5
  --
--  norm_infty (Vn v1) = realToFrac $ V.foldl max 0 v1

  -- | Generic p-norms. The usual norm in R^n is (norm_p 2).
  --
  --   Examples:
  --
  --   >>> let v1 = make2d (3,4)
  --   >>> norm_p 1 v1
  --   7.0
  --   >>> norm_p 2 v1
  --   5.0
  --
--  norm_p p (Vn v1) =
--    realToFrac $ root $ V.sum $ V.map (exponentiate . abs) v1
--    where
--      exponentiate = (** (fromIntegral p))
--      root = (** (recip (fromIntegral p)))





-- | Convenient constructor for 2D vectors.
--
--   Examples:
--
--   >>> import Roots.Simple
--   >>> let h = 0.5 :: Double
--   >>> let g1 (Vn (Vec2D x y)) = 1.0 + h*exp(-(x^2))/(1.0 + y^2)
--   >>> let g2 (Vn (Vec2D x y)) = 0.5 + h*atan(x^2 + y^2)
--   >>> let g u = make2d ((g1 u), (g2 u))
--   >>> let u0 = make2d (1.0, 1.0)
--   >>> let eps = 1/(10^9)
--   >>> fixed_point g eps u0
--   (1.0728549599342185,1.0820591495686167)
--