module FixedVector
where
-import Data.Vector.Fixed as V
-import Data.Vector.Fixed.Boxed
-import Data.Vector.Fixed.Internal
+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 (
+ foldl,
+ length,
+ map,
+ replicate,
+ sum,
+ zipWith
+ )
import Normed
-- | The Vn newtype simply wraps (Vector v a) so that we avoid
-- undecidable instances.
-newtype Vn a = Vn a
- deriving (Show)
+newtype Vn v a = Vn (v a)
+
+-- | Declare the dimension of the wrapper to be the dimension of what
+-- it contains.
+type instance Dim (Vn v) = Dim v
+
+instance (Vector v a) => Vector (Vn v) a where
+ -- | Fortunately, 'Fun' is an instance of 'Functor'. The
+ -- 'construct' defined on our contained type will return a
+ -- 'Fun', and we simply slap our constructor on top with fmap.
+ construct = fmap Vn construct
+
+ -- | Defer to the inspect defined on the contained type.
+ inspect (Vn v1) = inspect v1
+
+instance (Show a, Vector v a) => Show (Vn v a) where
+ -- | Display vectors as ordinary tuples. This is poor practice, but
+ -- these results are primarily displayed interactively and
+ -- convenience trumps correctness (said the guy who insists his
+ -- vector lengths be statically checked at compile-time).
+ --
+ -- Examples:
+ --
+ -- >>> let v1 = make2d (1,2)
+ -- >>> show v1
+ -- (1,2)
+ --
+ show (Vn v1) =
+ "(" ++ (intercalate "," element_strings) ++ ")"
+ where
+ v1l = toList v1
+ element_strings = Prelude.map show v1l
+
-- | We would really like to say, "anything that is a vector of
-- equatable things is itself equatable." The 'Vn' class
-- >>> v1 == v3
-- False
--
-instance (Eq a, Vector v a, Vector v Bool) => Eq (Vn (v a)) where
+instance (Eq a, Vector v a, Vector v Bool) => Eq (Vn v a) where
(Vn v1) == (Vn v2) = V.foldl (&&) True (V.zipWith (==) v1 v2)
+
-- | The use of 'Num' here is of course incorrect (otherwise, we
-- wouldn't have to throw errors). But it's really nice to be able
-- to use normal addition/subtraction.
-instance (Num a, Vector v a) => Num (Vn (v a)) where
+instance (Num a, Vector v a) => Num (Vn v a) where
-- | Componentwise addition.
--
-- Examples:
-- >>> let v1 = make2d (1,2)
-- >>> let v2 = make2d (3,4)
-- >>> v1 + v2
- -- Vn fromList [4,6]
+ -- (4,6)
--
(Vn v1) + (Vn v2) = Vn $ V.zipWith (+) v1 v2
+ -- | Componentwise subtraction.
+ --
+ -- Examples:
+ --
+ -- >>> let v1 = make2d (1,2)
+ -- >>> let v2 = make2d (3,4)
+ -- >>> v1 - v2
+ -- (-2,-2)
+ --
(Vn v1) - (Vn v2) = Vn $ V.zipWith (-) v1 v2
+
+ -- | Create an n-vector whose components are all equal to the given
+ -- integer. The result type must be specified since otherwise the
+ -- length n would be unknown.
+ --
+ -- Examples:
+ --
+ -- >>> let v1 = fromInteger 17 :: Vn Vec3 Int
+ -- (17,17,17)
+ --
fromInteger x = Vn $ V.replicate (fromInteger x)
(*) = error "multiplication of vectors is undefined"
abs = error "absolute value of vectors is undefined"
signum = error "signum of vectors is undefined"
-instance Functor Vn where
- fmap f (Vn v1) = Vn (f v1)
-instance (RealFloat a, Ord a, Vector v a) => Normed (Vn (v a)) where
- -- 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.
+-- | This is probably useless, since the vectors we usually contain
+-- aren't functor instances.
+instance (Functor v) => Functor (Vn v) where
+ fmap f (Vn v1) = Vn (f `fmap` v1)
+
+
+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) = fromRational $ toRational $ 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) =
fromRational $ toRational $ root $ V.sum $ V.map (exponentiate . abs) v1
where
root = (** (recip (fromIntegral p)))
-- | Dot (standard inner) product.
-dot :: (Num a, Vector v a) => Vn (v a) -> Vn (v a) -> a
+--
+-- Examples:
+--
+-- >>> let v1 = make3d (1,2,3)
+-- >>> let v2 = make3d (4,5,6)
+-- >>> dot v1 v2
+-- 32
+--
+dot :: (Num a, Vector v a) => Vn v a -> Vn v a -> a
dot (Vn v1) (Vn v2) = V.sum $ V.zipWith (*) v1 v2
+
-- | The angle between @v1@ and @v2@ in Euclidean space.
-angle :: (RealFloat a, Vector v a) => Vn (v a) -> Vn (v a) -> a
+--
+-- Examples:
+--
+-- >>> let v1 = make2d (1.0, 0.0)
+-- >>> let v2 = make2d (0.0, 1.0)
+-- >>> angle v1 v2 == pi/2.0
+-- True
+--
+angle :: (RealFloat a, Vector v a) => Vn v a -> Vn v a -> a
angle v1 v2 =
acos theta
where
theta = (v1 `dot` v2) / norms
- norms = (norm_p 2 v1) * (norm_p 2 v2)
+ norms = (norm v1) * (norm v2)
--- | Convenience function for 2d vectors.
-make2d :: forall a. (a,a) -> Vn (Vec2 a)
-make2d (x,y) =
- Vn v1
- where
- v1 = vec $ con |> x |> y :: Vec2 a
--- | Convenience function for 3d vectors.
-make3d :: forall a. (a,a,a) -> Vn (Vec3 a)
-make3d (x,y,z) =
- Vn v1
- where
- v1 = vec $ con |> x |> y |> z :: Vec3 a
+-- | Safe indexing.
+--
+-- Examples:
+--
+-- >>> let v1 = make3d (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
+
+
+
+
+-- * 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 Vec2D a = Vec2D a a
+type instance Dim Vec2D = N2
+instance Vector Vec2D a where
+ inspect (Vec2D x y) (Fun f) = f x y
+ construct = Fun Vec2D
+
+data Vec3D a = Vec3D a a a
+type instance Dim Vec3D = N3
+instance Vector Vec3D a where
+ inspect (Vec3D x y z) (Fun f) = f x y z
+ construct = Fun Vec3D
+
+data Vec4D a = Vec4D a a a a
+type instance Dim Vec4D = N4
+instance Vector Vec4D a where
+ inspect (Vec4D w x y z) (Fun f) = f w x y z
+ construct = Fun Vec4D
+
+
+-- | Convenience function for creating 2d vectors.
+--
+-- Examples:
+--
+-- >>> let v1 = make2d (1,2)
+-- >>> v1
+-- (1,2)
+-- >>> let Vn (Vec2D x y) = v1
+-- >>> (x,y)
+-- (1,2)
+--
+make2d :: forall a. (a,a) -> Vn Vec2D a
+make2d (x,y) = Vn (Vec2D x y)
+
+
+-- | Convenience function for creating 3d vectors.
+--
+-- Examples:
+--
+-- >>> let v1 = make3d (1,2,3)
+-- >>> v1
+-- (1,2,3)
+-- >>> let Vn (Vec3D x y z) = v1
+-- >>> (x,y,z)
+-- (1,2,3)
+--
+make3d :: forall a. (a,a,a) -> Vn Vec3D a
+make3d (x,y,z) = Vn (Vec3D x y z)
+
+
+-- | Convenience function for creating 4d vectors.
+--
+-- Examples:
+--
+-- >>> let v1 = make4d (1,2,3,4)
+-- >>> v1
+-- (1,2,3,4)
+-- >>> let Vn (Vec4D w x y z) = v1
+-- >>> (w,x,y,z)
+-- (1,2,3,4)
+--
+make4d :: forall a. (a,a,a,a) -> Vn Vec4D a
+make4d (w,x,y,z) = Vn (Vec4D w x y z)
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