linear-1.20.7: Linear Algebra

Linear.Vector

Description

Operations on free vector spaces.

Synopsis

# Documentation

class Functor f => Additive f where Source #

Methods

zero :: Num a => f a Source #

The zero vector

zero :: (GAdditive (Rep1 f), Generic1 f, Num a) => f a Source #

The zero vector

(^+^) :: Num a => f a -> f a -> f a infixl 6 Source #

Compute the sum of two vectors

>>> V2 1 2 ^+^ V2 3 4
V2 4 6


(^-^) :: Num a => f a -> f a -> f a infixl 6 Source #

Compute the difference between two vectors

>>> V2 4 5 ^-^ V2 3 1
V2 1 4


lerp :: Num a => a -> f a -> f a -> f a Source #

Linearly interpolate between two vectors.

liftU2 :: (a -> a -> a) -> f a -> f a -> f a Source #

Apply a function to merge the 'non-zero' components of two vectors, unioning the rest of the values.

• For a dense vector this is equivalent to liftA2.
• For a sparse vector this is equivalent to unionWith.

liftU2 :: Applicative f => (a -> a -> a) -> f a -> f a -> f a Source #

Apply a function to merge the 'non-zero' components of two vectors, unioning the rest of the values.

• For a dense vector this is equivalent to liftA2.
• For a sparse vector this is equivalent to unionWith.

liftI2 :: (a -> b -> c) -> f a -> f b -> f c Source #

Apply a function to the components of two vectors.

• For a dense vector this is equivalent to liftA2.
• For a sparse vector this is equivalent to intersectionWith.

liftI2 :: Applicative f => (a -> b -> c) -> f a -> f b -> f c Source #

Apply a function to the components of two vectors.

• For a dense vector this is equivalent to liftA2.
• For a sparse vector this is equivalent to intersectionWith.

Instances

 Additive [] Source # Methodszero :: Num a => [a] Source #(^+^) :: Num a => [a] -> [a] -> [a] Source #(^-^) :: Num a => [a] -> [a] -> [a] Source #lerp :: Num a => a -> [a] -> [a] -> [a] Source #liftU2 :: (a -> a -> a) -> [a] -> [a] -> [a] Source #liftI2 :: (a -> b -> c) -> [a] -> [b] -> [c] Source # Source # Methodszero :: Num a => Maybe a Source #(^+^) :: Num a => Maybe a -> Maybe a -> Maybe a Source #(^-^) :: Num a => Maybe a -> Maybe a -> Maybe a Source #lerp :: Num a => a -> Maybe a -> Maybe a -> Maybe a Source #liftU2 :: (a -> a -> a) -> Maybe a -> Maybe a -> Maybe a Source #liftI2 :: (a -> b -> c) -> Maybe a -> Maybe b -> Maybe c Source # Source # Methodszero :: Num a => Identity a Source #(^+^) :: Num a => Identity a -> Identity a -> Identity a Source #(^-^) :: Num a => Identity a -> Identity a -> Identity a Source #lerp :: Num a => a -> Identity a -> Identity a -> Identity a Source #liftU2 :: (a -> a -> a) -> Identity a -> Identity a -> Identity a Source #liftI2 :: (a -> b -> c) -> Identity a -> Identity b -> Identity c Source # Source # Methodszero :: Num a => Complex a Source #(^+^) :: Num a => Complex a -> Complex a -> Complex a Source #(^-^) :: Num a => Complex a -> Complex a -> Complex a Source #lerp :: Num a => a -> Complex a -> Complex a -> Complex a Source #liftU2 :: (a -> a -> a) -> Complex a -> Complex a -> Complex a Source #liftI2 :: (a -> b -> c) -> Complex a -> Complex b -> Complex c Source # Source # Methodszero :: Num a => ZipList a Source #(^+^) :: Num a => ZipList a -> ZipList a -> ZipList a Source #(^-^) :: Num a => ZipList a -> ZipList a -> ZipList a Source #lerp :: Num a => a -> ZipList a -> ZipList a -> ZipList a Source #liftU2 :: (a -> a -> a) -> ZipList a -> ZipList a -> ZipList a Source #liftI2 :: (a -> b -> c) -> ZipList a -> ZipList b -> ZipList c Source # Source # Methodszero :: Num a => IntMap a Source #(^+^) :: Num a => IntMap a -> IntMap a -> IntMap a Source #(^-^) :: Num a => IntMap a -> IntMap a -> IntMap a Source #lerp :: Num a => a -> IntMap a -> IntMap a -> IntMap a Source #liftU2 :: (a -> a -> a) -> IntMap a -> IntMap a -> IntMap a Source #liftI2 :: (a -> b -> c) -> IntMap a -> IntMap b -> IntMap c Source # Source # Methodszero :: Num a => Vector a Source #(^+^) :: Num a => Vector a -> Vector a -> Vector a Source #(^-^) :: Num a => Vector a -> Vector a -> Vector a Source #lerp :: Num a => a -> Vector a -> Vector a -> Vector a Source #liftU2 :: (a -> a -> a) -> Vector a -> Vector a -> Vector a Source #liftI2 :: (a -> b -> c) -> Vector a -> Vector b -> Vector c Source # Source # Methodszero :: Num a => V0 a Source #(^+^) :: Num a => V0 a -> V0 a -> V0 a Source #(^-^) :: Num a => V0 a -> V0 a -> V0 a Source #lerp :: Num a => a -> V0 a -> V0 a -> V0 a Source #liftU2 :: (a -> a -> a) -> V0 a -> V0 a -> V0 a Source #liftI2 :: (a -> b -> c) -> V0 a -> V0 b -> V0 c Source # Source # Methodszero :: Num a => V1 a Source #(^+^) :: Num a => V1 a -> V1 a -> V1 a Source #(^-^) :: Num a => V1 a -> V1 a -> V1 a Source #lerp :: Num a => a -> V1 a -> V1 a -> V1 a Source #liftU2 :: (a -> a -> a) -> V1 a -> V1 a -> V1 a Source #liftI2 :: (a -> b -> c) -> V1 a -> V1 b -> V1 c Source # Source # Methodszero :: Num a => V2 a Source #(^+^) :: Num a => V2 a -> V2 a -> V2 a Source #(^-^) :: Num a => V2 a -> V2 a -> V2 a Source #lerp :: Num a => a -> V2 a -> V2 a -> V2 a Source #liftU2 :: (a -> a -> a) -> V2 a -> V2 a -> V2 a Source #liftI2 :: (a -> b -> c) -> V2 a -> V2 b -> V2 c Source # Source # Methodszero :: Num a => V3 a Source #(^+^) :: Num a => V3 a -> V3 a -> V3 a Source #(^-^) :: Num a => V3 a -> V3 a -> V3 a Source #lerp :: Num a => a -> V3 a -> V3 a -> V3 a Source #liftU2 :: (a -> a -> a) -> V3 a -> V3 a -> V3 a Source #liftI2 :: (a -> b -> c) -> V3 a -> V3 b -> V3 c Source # Source # Methodszero :: Num a => V4 a Source #(^+^) :: Num a => V4 a -> V4 a -> V4 a Source #(^-^) :: Num a => V4 a -> V4 a -> V4 a Source #lerp :: Num a => a -> V4 a -> V4 a -> V4 a Source #liftU2 :: (a -> a -> a) -> V4 a -> V4 a -> V4 a Source #liftI2 :: (a -> b -> c) -> V4 a -> V4 b -> V4 c Source # Source # Methodszero :: Num a => Plucker a Source #(^+^) :: Num a => Plucker a -> Plucker a -> Plucker a Source #(^-^) :: Num a => Plucker a -> Plucker a -> Plucker a Source #lerp :: Num a => a -> Plucker a -> Plucker a -> Plucker a Source #liftU2 :: (a -> a -> a) -> Plucker a -> Plucker a -> Plucker a Source #liftI2 :: (a -> b -> c) -> Plucker a -> Plucker b -> Plucker c Source # Source # Methodszero :: Num a => Quaternion a Source #(^+^) :: Num a => Quaternion a -> Quaternion a -> Quaternion a Source #(^-^) :: Num a => Quaternion a -> Quaternion a -> Quaternion a Source #lerp :: Num a => a -> Quaternion a -> Quaternion a -> Quaternion a Source #liftU2 :: (a -> a -> a) -> Quaternion a -> Quaternion a -> Quaternion a Source #liftI2 :: (a -> b -> c) -> Quaternion a -> Quaternion b -> Quaternion c Source # Additive ((->) b) Source # Methodszero :: Num a => b -> a Source #(^+^) :: Num a => (b -> a) -> (b -> a) -> b -> a Source #(^-^) :: Num a => (b -> a) -> (b -> a) -> b -> a Source #lerp :: Num a => a -> (b -> a) -> (b -> a) -> b -> a Source #liftU2 :: (a -> a -> a) -> (b -> a) -> (b -> a) -> b -> a Source #liftI2 :: (a -> b -> c) -> (b -> a) -> (b -> b) -> b -> c Source # Ord k => Additive (Map k) Source # Methodszero :: Num a => Map k a Source #(^+^) :: Num a => Map k a -> Map k a -> Map k a Source #(^-^) :: Num a => Map k a -> Map k a -> Map k a Source #lerp :: Num a => a -> Map k a -> Map k a -> Map k a Source #liftU2 :: (a -> a -> a) -> Map k a -> Map k a -> Map k a Source #liftI2 :: (a -> b -> c) -> Map k a -> Map k b -> Map k c Source # (Eq k, Hashable k) => Additive (HashMap k) Source # Methodszero :: Num a => HashMap k a Source #(^+^) :: Num a => HashMap k a -> HashMap k a -> HashMap k a Source #(^-^) :: Num a => HashMap k a -> HashMap k a -> HashMap k a Source #lerp :: Num a => a -> HashMap k a -> HashMap k a -> HashMap k a Source #liftU2 :: (a -> a -> a) -> HashMap k a -> HashMap k a -> HashMap k a Source #liftI2 :: (a -> b -> c) -> HashMap k a -> HashMap k b -> HashMap k c Source # Additive f => Additive (Point f) Source # Methodszero :: Num a => Point f a Source #(^+^) :: Num a => Point f a -> Point f a -> Point f a Source #(^-^) :: Num a => Point f a -> Point f a -> Point f a Source #lerp :: Num a => a -> Point f a -> Point f a -> Point f a Source #liftU2 :: (a -> a -> a) -> Point f a -> Point f a -> Point f a Source #liftI2 :: (a -> b -> c) -> Point f a -> Point f b -> Point f c Source # Dim k n => Additive (V k n) Source # Methodszero :: Num a => V k n a Source #(^+^) :: Num a => V k n a -> V k n a -> V k n a Source #(^-^) :: Num a => V k n a -> V k n a -> V k n a Source #lerp :: Num a => a -> V k n a -> V k n a -> V k n a Source #liftU2 :: (a -> a -> a) -> V k n a -> V k n a -> V k n a Source #liftI2 :: (a -> b -> c) -> V k n a -> V k n b -> V k n c Source #

newtype E t Source #

Basis element

Constructors

 E Fieldsel :: forall x. Lens' (t x) x

Instances

 (Num r, TrivialConjugate r) => Coalgebra r (E Quaternion) Source # Methodscomult :: (E Quaternion -> r) -> E Quaternion -> E Quaternion -> r Source #counital :: (E Quaternion -> r) -> r Source # Num r => Coalgebra r (E Complex) Source # Methodscomult :: (E Complex -> r) -> E Complex -> E Complex -> r Source #counital :: (E Complex -> r) -> r Source # Num r => Coalgebra r (E V4) Source # Methodscomult :: (E V4 -> r) -> E V4 -> E V4 -> r Source #counital :: (E V4 -> r) -> r Source # Num r => Coalgebra r (E V3) Source # Methodscomult :: (E V3 -> r) -> E V3 -> E V3 -> r Source #counital :: (E V3 -> r) -> r Source # Num r => Coalgebra r (E V2) Source # Methodscomult :: (E V2 -> r) -> E V2 -> E V2 -> r Source #counital :: (E V2 -> r) -> r Source # Num r => Coalgebra r (E V1) Source # Methodscomult :: (E V1 -> r) -> E V1 -> E V1 -> r Source #counital :: (E V1 -> r) -> r Source # Num r => Coalgebra r (E V0) Source # Methodscomult :: (E V0 -> r) -> E V0 -> E V0 -> r Source #counital :: (E V0 -> r) -> r Source # (Num r, TrivialConjugate r) => Algebra r (E Quaternion) Source # Methodsmult :: (E Quaternion -> E Quaternion -> r) -> E Quaternion -> r Source #unital :: r -> E Quaternion -> r Source # Num r => Algebra r (E Complex) Source # Methodsmult :: (E Complex -> E Complex -> r) -> E Complex -> r Source #unital :: r -> E Complex -> r Source # Num r => Algebra r (E V1) Source # Methodsmult :: (E V1 -> E V1 -> r) -> E V1 -> r Source #unital :: r -> E V1 -> r Source # Num r => Algebra r (E V0) Source # Methodsmult :: (E V0 -> E V0 -> r) -> E V0 -> r Source #unital :: r -> E V0 -> r Source # # Methodsimap :: (E V0 -> a -> b) -> V0 a -> V0 b #imapped :: (Indexable (E V0) p, Settable f) => p a (f b) -> V0 a -> f (V0 b) # # Methodsimap :: (E V1 -> a -> b) -> V1 a -> V1 b #imapped :: (Indexable (E V1) p, Settable f) => p a (f b) -> V1 a -> f (V1 b) # # Methodsimap :: (E V2 -> a -> b) -> V2 a -> V2 b #imapped :: (Indexable (E V2) p, Settable f) => p a (f b) -> V2 a -> f (V2 b) # # Methodsimap :: (E V3 -> a -> b) -> V3 a -> V3 b #imapped :: (Indexable (E V3) p, Settable f) => p a (f b) -> V3 a -> f (V3 b) # # Methodsimap :: (E V4 -> a -> b) -> V4 a -> V4 b #imapped :: (Indexable (E V4) p, Settable f) => p a (f b) -> V4 a -> f (V4 b) # # Methodsimap :: (E Plucker -> a -> b) -> Plucker a -> Plucker b #imapped :: (Indexable (E Plucker) p, Settable f) => p a (f b) -> Plucker a -> f (Plucker b) # # Methodsimap :: (E Quaternion -> a -> b) -> Quaternion a -> Quaternion b #imapped :: (Indexable (E Quaternion) p, Settable f) => p a (f b) -> Quaternion a -> f (Quaternion b) # # MethodsifoldMap :: Monoid m => (E V0 -> a -> m) -> V0 a -> m #ifolded :: (Indexable (E V0) p, Contravariant f, Applicative f) => p a (f a) -> V0 a -> f (V0 a) #ifoldr :: (E V0 -> a -> b -> b) -> b -> V0 a -> b #ifoldl :: (E V0 -> b -> a -> b) -> b -> V0 a -> b #ifoldr' :: (E V0 -> a -> b -> b) -> b -> V0 a -> b #ifoldl' :: (E V0 -> b -> a -> b) -> b -> V0 a -> b # # MethodsifoldMap :: Monoid m => (E V1 -> a -> m) -> V1 a -> m #ifolded :: (Indexable (E V1) p, Contravariant f, Applicative f) => p a (f a) -> V1 a -> f (V1 a) #ifoldr :: (E V1 -> a -> b -> b) -> b -> V1 a -> b #ifoldl :: (E V1 -> b -> a -> b) -> b -> V1 a -> b #ifoldr' :: (E V1 -> a -> b -> b) -> b -> V1 a -> b #ifoldl' :: (E V1 -> b -> a -> b) -> b -> V1 a -> b # # MethodsifoldMap :: Monoid m => (E V2 -> a -> m) -> V2 a -> m #ifolded :: (Indexable (E V2) p, Contravariant f, Applicative f) => p a (f a) -> V2 a -> f (V2 a) #ifoldr :: (E V2 -> a -> b -> b) -> b -> V2 a -> b #ifoldl :: (E V2 -> b -> a -> b) -> b -> V2 a -> b #ifoldr' :: (E V2 -> a -> b -> b) -> b -> V2 a -> b #ifoldl' :: (E V2 -> b -> a -> b) -> b -> V2 a -> b # # MethodsifoldMap :: Monoid m => (E V3 -> a -> m) -> V3 a -> m #ifolded :: (Indexable (E V3) p, Contravariant f, Applicative f) => p a (f a) -> V3 a -> f (V3 a) #ifoldr :: (E V3 -> a -> b -> b) -> b -> V3 a -> b #ifoldl :: (E V3 -> b -> a -> b) -> b -> V3 a -> b #ifoldr' :: (E V3 -> a -> b -> b) -> b -> V3 a -> b #ifoldl' :: (E V3 -> b -> a -> b) -> b -> V3 a -> b # # MethodsifoldMap :: Monoid m => (E V4 -> a -> m) -> V4 a -> m #ifolded :: (Indexable (E V4) p, Contravariant f, Applicative f) => p a (f a) -> V4 a -> f (V4 a) #ifoldr :: (E V4 -> a -> b -> b) -> b -> V4 a -> b #ifoldl :: (E V4 -> b -> a -> b) -> b -> V4 a -> b #ifoldr' :: (E V4 -> a -> b -> b) -> b -> V4 a -> b #ifoldl' :: (E V4 -> b -> a -> b) -> b -> V4 a -> b # # MethodsifoldMap :: Monoid m => (E Plucker -> a -> m) -> Plucker a -> m #ifolded :: (Indexable (E Plucker) p, Contravariant f, Applicative f) => p a (f a) -> Plucker a -> f (Plucker a) #ifoldr :: (E Plucker -> a -> b -> b) -> b -> Plucker a -> b #ifoldl :: (E Plucker -> b -> a -> b) -> b -> Plucker a -> b #ifoldr' :: (E Plucker -> a -> b -> b) -> b -> Plucker a -> b #ifoldl' :: (E Plucker -> b -> a -> b) -> b -> Plucker a -> b # # MethodsifoldMap :: Monoid m => (E Quaternion -> a -> m) -> Quaternion a -> m #ifolded :: (Indexable (E Quaternion) p, Contravariant f, Applicative f) => p a (f a) -> Quaternion a -> f (Quaternion a) #ifoldr :: (E Quaternion -> a -> b -> b) -> b -> Quaternion a -> b #ifoldl :: (E Quaternion -> b -> a -> b) -> b -> Quaternion a -> b #ifoldr' :: (E Quaternion -> a -> b -> b) -> b -> Quaternion a -> b #ifoldl' :: (E Quaternion -> b -> a -> b) -> b -> Quaternion a -> b # # Methodsitraverse :: Applicative f => (E V0 -> a -> f b) -> V0 a -> f (V0 b) #itraversed :: (Indexable (E V0) p, Applicative f) => p a (f b) -> V0 a -> f (V0 b) # # Methodsitraverse :: Applicative f => (E V1 -> a -> f b) -> V1 a -> f (V1 b) #itraversed :: (Indexable (E V1) p, Applicative f) => p a (f b) -> V1 a -> f (V1 b) # # Methodsitraverse :: Applicative f => (E V2 -> a -> f b) -> V2 a -> f (V2 b) #itraversed :: (Indexable (E V2) p, Applicative f) => p a (f b) -> V2 a -> f (V2 b) # # Methodsitraverse :: Applicative f => (E V3 -> a -> f b) -> V3 a -> f (V3 b) #itraversed :: (Indexable (E V3) p, Applicative f) => p a (f b) -> V3 a -> f (V3 b) # # Methodsitraverse :: Applicative f => (E V4 -> a -> f b) -> V4 a -> f (V4 b) #itraversed :: (Indexable (E V4) p, Applicative f) => p a (f b) -> V4 a -> f (V4 b) # # Methodsitraverse :: Applicative f => (E Plucker -> a -> f b) -> Plucker a -> f (Plucker b) #itraversed :: (Indexable (E Plucker) p, Applicative f) => p a (f b) -> Plucker a -> f (Plucker b) # # Methodsitraverse :: Applicative f => (E Quaternion -> a -> f b) -> Quaternion a -> f (Quaternion b) #itraversed :: (Indexable (E Quaternion) p, Applicative f) => p a (f b) -> Quaternion a -> f (Quaternion b) #

negated :: (Functor f, Num a) => f a -> f a Source #

Compute the negation of a vector

>>> negated (V2 2 4)
V2 (-2) (-4)


(^*) :: (Functor f, Num a) => f a -> a -> f a infixl 7 Source #

Compute the right scalar product

>>> V2 3 4 ^* 2
V2 6 8


(*^) :: (Functor f, Num a) => a -> f a -> f a infixl 7 Source #

Compute the left scalar product

>>> 2 *^ V2 3 4
V2 6 8


(^/) :: (Functor f, Fractional a) => f a -> a -> f a infixl 7 Source #

Compute division by a scalar on the right.

sumV :: (Foldable f, Additive v, Num a) => f (v a) -> v a Source #

Sum over multiple vectors

>>> sumV [V2 1 1, V2 3 4]
V2 4 5


basis :: (Additive t, Traversable t, Num a) => [t a] Source #

Produce a default basis for a vector space. If the dimensionality of the vector space is not statically known, see basisFor.

basisFor :: (Traversable t, Num a) => t b -> [t a] Source #

Produce a default basis for a vector space from which the argument is drawn.

scaled :: (Traversable t, Num a) => t a -> t (t a) Source #

Produce a diagonal (scale) matrix from a vector.

>>> scaled (V2 2 3)
V2 (V2 2 0) (V2 0 3)


outer :: (Functor f, Functor g, Num a) => f a -> g a -> f (g a) Source #

Outer (tensor) product of two vectors

unit :: (Additive t, Num a) => ASetter' (t a) a -> t a Source #

Create a unit vector.

>>> unit _x :: V2 Int
V2 1 0