import Data.Vector.Fixed (
Dim,
Fun(..),
+ N1,
N2,
N3,
N4,
toList,
)
import qualified Data.Vector.Fixed as V (
- eq,
- foldl,
length,
- map,
- replicate,
- sum,
- zipWith
)
import Normed
--- | The Vn newtype simply wraps (Vector v a) so that we avoid
--- undecidable instances.
-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
+-- * 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.
-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
+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
- -- | Defer to the inspect defined on the contained type.
- inspect (Vn v1) = inspect v1
+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
-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
+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
--- | We would really like to say, "anything that is a vector of
--- equatable things is itself equatable." The 'Vn' class
--- allows us to express this without a GHC battle.
+-- | Unsafe indexing.
--
-- Examples:
--
--- >>> let v1 = make2d (1,2)
--- >>> let v2 = make2d (1,2)
--- >>> let v3 = make2d (3,4)
--- >>> v1 == v2
--- True
--- >>> v1 == v3
--- False
+-- >>> let v1 = Vec2D 1 2
+-- >>> v1 ! 1
+-- 2
--
-instance (Eq a, Vector v a) => Eq (Vn v a) where
- (Vn v1) == (Vn v2) = v1 `V.eq` 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
- -- | Componentwise addition.
- --
- -- Examples:
- --
- -- >>> let v1 = make2d (1,2)
- -- >>> let v2 = make2d (3,4)
- -- >>> v1 + v2
- -- (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
+(!) :: (Vector v a) => v a -> Int -> a
+(!) v1 idx = (toList v1) !! idx
- -- | 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"
+-- | 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
--- | 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
+--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.
-- >>> norm_infty v1
-- 5
--
- norm_infty (Vn v1) = realToFrac $ V.foldl max 0 v1
+-- norm_infty (Vn v1) = realToFrac $ V.foldl max 0 v1
-- | Generic p-norms. The usual norm in R^n is (norm_p 2).
--
-- >>> 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)))
-
--- | Dot (standard inner) product.
---
--- 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.
---
--- 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 v1) * (norm v2)
+-- norm_p p (Vn v1) =
+-- realToFrac $ root $ V.sum $ V.map (exponentiate . abs) v1
+-- where
+-- exponentiate = (** (fromIntegral p))
+-- root = (** (recip (fromIntegral p)))
--- | Unsafe indexing.
---
--- Examples:
---
--- >>> let v1 = make2d (1,2)
--- >>> v1 ! 1
--- 2
---
-(!) :: (Vector v a) => v a -> Int -> a
-(!) v1 idx = (toList v1) !! idx
--- | 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.
-
-- | Convenient constructor for 2D vectors.
--
-- Examples:
-- >>> fixed_point g eps u0
-- (1.0728549599342185,1.0820591495686167)
--
-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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