--- | 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
--- allows us to express this without a GHC battle.
---
--- Examples:
---
--- >>> let v1 = make2d (1,2)
--- >>> let v2 = make2d (1,2)
--- >>> let v3 = make2d (3,4)
--- >>> v1 == v2
--- True
--- >>> v1 == v3
--- False
---
-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
-
- -- | 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"
-
-
--- | 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) = 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)))
-
--- | 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)
-
-
--- | Unsafe indexing.
---
--- Examples:
---
--- >>> let v1 = make2d (1,2)
--- >>> v1 ! 1
--- 2
---
-(!) :: (Vector v a) => v a -> Int -> a
-(!) v1 idx = (toList v1) !! idx