- -- | 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)