Linear.Vector

Operations on free vector spaces.

Documentation

class Functor f => Additive f where

A vector is an additive group with additional structure.

Minimal complete definition

Nothing

Methods

zero :: Num a => f a

The zero vector

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

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

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

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

Linearly interpolate between two vectors.

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

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.

default liftU2 :: Applicative f => (a -> a -> a) -> f a -> f a -> f a

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

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.

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

Instances

Additive []

Defined in Linear.Vector

Methods

zero :: Num a => [a]

(^+^) :: Num a => [a] -> [a] -> [a]

(^-^) :: Num a => [a] -> [a] -> [a]

lerp :: Num a => a -> [a] -> [a] -> [a]

liftU2 :: (a -> a -> a) -> [a] -> [a] -> [a]

liftI2 :: (a -> b -> c) -> [a] -> [b] -> [c]

Additive Maybe

Defined in Linear.Vector

Methods

zero :: Num a => Maybe a

(^+^) :: Num a => Maybe a -> Maybe a -> Maybe a

(^-^) :: Num a => Maybe a -> Maybe a -> Maybe a

lerp :: Num a => a -> Maybe a -> Maybe a -> Maybe a

liftU2 :: (a -> a -> a) -> Maybe a -> Maybe a -> Maybe a

liftI2 :: (a -> b -> c) -> Maybe a -> Maybe b -> Maybe c

Additive Complex

Defined in Linear.Vector

Methods

zero :: Num a => Complex a

(^+^) :: Num a => Complex a -> Complex a -> Complex a

(^-^) :: Num a => Complex a -> Complex a -> Complex a

lerp :: Num a => a -> Complex a -> Complex a -> Complex a

liftU2 :: (a -> a -> a) -> Complex a -> Complex a -> Complex a

liftI2 :: (a -> b -> c) -> Complex a -> Complex b -> Complex c

Additive ZipList

Defined in Linear.Vector

Methods

zero :: Num a => ZipList a

(^+^) :: Num a => ZipList a -> ZipList a -> ZipList a

(^-^) :: Num a => ZipList a -> ZipList a -> ZipList a

lerp :: Num a => a -> ZipList a -> ZipList a -> ZipList a

liftU2 :: (a -> a -> a) -> ZipList a -> ZipList a -> ZipList a

liftI2 :: (a -> b -> c) -> ZipList a -> ZipList b -> ZipList c

Additive Identity

Defined in Linear.Vector

Methods

zero :: Num a => Identity a

(^+^) :: Num a => Identity a -> Identity a -> Identity a

(^-^) :: Num a => Identity a -> Identity a -> Identity a

lerp :: Num a => a -> Identity a -> Identity a -> Identity a

liftU2 :: (a -> a -> a) -> Identity a -> Identity a -> Identity a

liftI2 :: (a -> b -> c) -> Identity a -> Identity b -> Identity c

Additive IntMap

Defined in Linear.Vector

Methods

zero :: Num a => IntMap a

(^+^) :: Num a => IntMap a -> IntMap a -> IntMap a

(^-^) :: Num a => IntMap a -> IntMap a -> IntMap a

lerp :: Num a => a -> IntMap a -> IntMap a -> IntMap a

liftU2 :: (a -> a -> a) -> IntMap a -> IntMap a -> IntMap a

liftI2 :: (a -> b -> c) -> IntMap a -> IntMap b -> IntMap c

Additive Vector

Defined in Linear.Vector

Methods

zero :: Num a => Vector a

(^+^) :: Num a => Vector a -> Vector a -> Vector a

(^-^) :: Num a => Vector a -> Vector a -> Vector a

lerp :: Num a => a -> Vector a -> Vector a -> Vector a

liftU2 :: (a -> a -> a) -> Vector a -> Vector a -> Vector a

liftI2 :: (a -> b -> c) -> Vector a -> Vector b -> Vector c

Additive V1

Defined in Linear.V1

Methods

zero :: Num a => V1 a

(^+^) :: Num a => V1 a -> V1 a -> V1 a

(^-^) :: Num a => V1 a -> V1 a -> V1 a

lerp :: Num a => a -> V1 a -> V1 a -> V1 a

liftU2 :: (a -> a -> a) -> V1 a -> V1 a -> V1 a

liftI2 :: (a -> b -> c) -> V1 a -> V1 b -> V1 c

Additive V2

Defined in Linear.V2

Methods

zero :: Num a => V2 a

(^+^) :: Num a => V2 a -> V2 a -> V2 a

(^-^) :: Num a => V2 a -> V2 a -> V2 a

lerp :: Num a => a -> V2 a -> V2 a -> V2 a

liftU2 :: (a -> a -> a) -> V2 a -> V2 a -> V2 a

liftI2 :: (a -> b -> c) -> V2 a -> V2 b -> V2 c

Additive V3

Defined in Linear.V3

Methods

zero :: Num a => V3 a

(^+^) :: Num a => V3 a -> V3 a -> V3 a

(^-^) :: Num a => V3 a -> V3 a -> V3 a

lerp :: Num a => a -> V3 a -> V3 a -> V3 a

liftU2 :: (a -> a -> a) -> V3 a -> V3 a -> V3 a

liftI2 :: (a -> b -> c) -> V3 a -> V3 b -> V3 c

Additive V4

Defined in Linear.V4

Methods

zero :: Num a => V4 a

(^+^) :: Num a => V4 a -> V4 a -> V4 a

(^-^) :: Num a => V4 a -> V4 a -> V4 a

lerp :: Num a => a -> V4 a -> V4 a -> V4 a

liftU2 :: (a -> a -> a) -> V4 a -> V4 a -> V4 a

liftI2 :: (a -> b -> c) -> V4 a -> V4 b -> V4 c

Additive V0

Defined in Linear.V0

Methods

zero :: Num a => V0 a

(^+^) :: Num a => V0 a -> V0 a -> V0 a

(^-^) :: Num a => V0 a -> V0 a -> V0 a

lerp :: Num a => a -> V0 a -> V0 a -> V0 a

liftU2 :: (a -> a -> a) -> V0 a -> V0 a -> V0 a

liftI2 :: (a -> b -> c) -> V0 a -> V0 b -> V0 c

Additive Quaternion

Defined in Linear.Quaternion

Methods

zero :: Num a => Quaternion a

(^+^) :: Num a => Quaternion a -> Quaternion a -> Quaternion a

(^-^) :: Num a => Quaternion a -> Quaternion a -> Quaternion a

lerp :: Num a => a -> Quaternion a -> Quaternion a -> Quaternion a

liftU2 :: (a -> a -> a) -> Quaternion a -> Quaternion a -> Quaternion a

liftI2 :: (a -> b -> c) -> Quaternion a -> Quaternion b -> Quaternion c

Additive Plucker

Defined in Linear.Plucker

Methods

zero :: Num a => Plucker a

(^+^) :: Num a => Plucker a -> Plucker a -> Plucker a

(^-^) :: Num a => Plucker a -> Plucker a -> Plucker a

lerp :: Num a => a -> Plucker a -> Plucker a -> Plucker a

liftU2 :: (a -> a -> a) -> Plucker a -> Plucker a -> Plucker a

liftI2 :: (a -> b -> c) -> Plucker a -> Plucker b -> Plucker c

Ord k => Additive (Map k)

Defined in Linear.Vector

Methods

zero :: Num a => Map k a

(^+^) :: Num a => Map k a -> Map k a -> Map k a

(^-^) :: Num a => Map k a -> Map k a -> Map k a

lerp :: Num a => a -> Map k a -> Map k a -> Map k a

liftU2 :: (a -> a -> a) -> Map k a -> Map k a -> Map k a

liftI2 :: (a -> b -> c) -> Map k a -> Map k b -> Map k c

(Eq k, Hashable k) => Additive (HashMap k)

Defined in Linear.Vector

Methods

zero :: Num a => HashMap k a

(^+^) :: Num a => HashMap k a -> HashMap k a -> HashMap k a

(^-^) :: Num a => HashMap k a -> HashMap k a -> HashMap k a

lerp :: Num a => a -> HashMap k a -> HashMap k a -> HashMap k a

liftU2 :: (a -> a -> a) -> HashMap k a -> HashMap k a -> HashMap k a

liftI2 :: (a -> b -> c) -> HashMap k a -> HashMap k b -> HashMap k c

Additive f => Additive (Point f)

Defined in Linear.Affine

Methods

zero :: Num a => Point f a

(^+^) :: Num a => Point f a -> Point f a -> Point f a

(^-^) :: Num a => Point f a -> Point f a -> Point f a

lerp :: Num a => a -> Point f a -> Point f a -> Point f a

liftU2 :: (a -> a -> a) -> Point f a -> Point f a -> Point f a

liftI2 :: (a -> b -> c) -> Point f a -> Point f b -> Point f c

Dim n => Additive (V n)

Defined in Linear.V

Methods

zero :: Num a => V n a

(^+^) :: Num a => V n a -> V n a -> V n a

(^-^) :: Num a => V n a -> V n a -> V n a

lerp :: Num a => a -> V n a -> V n a -> V n a

liftU2 :: (a -> a -> a) -> V n a -> V n a -> V n a

liftI2 :: (a -> b -> c) -> V n a -> V n b -> V n c

Additive ((->) b :: Type -> Type)

Defined in Linear.Vector

Methods

zero :: Num a => b -> a

(^+^) :: Num a => (b -> a) -> (b -> a) -> b -> a

(^-^) :: Num a => (b -> a) -> (b -> a) -> b -> a

lerp :: Num a => a -> (b -> a) -> (b -> a) -> b -> a

liftU2 :: (a -> a -> a) -> (b -> a) -> (b -> a) -> b -> a

liftI2 :: (a -> b0 -> c) -> (b -> a) -> (b -> b0) -> b -> c

(Additive f, Additive g) => Additive (Product f g)

Defined in Linear.Vector

Methods

zero :: Num a => Product f g a

(^+^) :: Num a => Product f g a -> Product f g a -> Product f g a

(^-^) :: Num a => Product f g a -> Product f g a -> Product f g a

lerp :: Num a => a -> Product f g a -> Product f g a -> Product f g a

liftU2 :: (a -> a -> a) -> Product f g a -> Product f g a -> Product f g a

liftI2 :: (a -> b -> c) -> Product f g a -> Product f g b -> Product f g c

(Additive f, Additive g) => Additive (Compose f g)

Defined in Linear.Vector

Methods

zero :: Num a => Compose f g a

(^+^) :: Num a => Compose f g a -> Compose f g a -> Compose f g a

(^-^) :: Num a => Compose f g a -> Compose f g a -> Compose f g a

lerp :: Num a => a -> Compose f g a -> Compose f g a -> Compose f g a

liftU2 :: (a -> a -> a) -> Compose f g a -> Compose f g a -> Compose f g a

liftI2 :: (a -> b -> c) -> Compose f g a -> Compose f g b -> Compose f g c

newtype E t

Basis element

Constructors

E

Fields

• el :: forall x. Lens' (t x) x

Instances

(Num r, TrivialConjugate r) => Coalgebra r (E Quaternion)

Defined in Linear.Algebra

Methods

comult :: (E Quaternion -> r) -> E Quaternion -> E Quaternion -> r

counital :: (E Quaternion -> r) -> r

Num r => Coalgebra r (E Complex)

Defined in Linear.Algebra

Methods

comult :: (E Complex -> r) -> E Complex -> E Complex -> r

counital :: (E Complex -> r) -> r

Num r => Coalgebra r (E V4)

Defined in Linear.Algebra

Methods

comult :: (E V4 -> r) -> E V4 -> E V4 -> r

counital :: (E V4 -> r) -> r

Num r => Coalgebra r (E V3)

Defined in Linear.Algebra

Methods

comult :: (E V3 -> r) -> E V3 -> E V3 -> r

counital :: (E V3 -> r) -> r

Num r => Coalgebra r (E V2)

Defined in Linear.Algebra

Methods

comult :: (E V2 -> r) -> E V2 -> E V2 -> r

counital :: (E V2 -> r) -> r

Num r => Coalgebra r (E V1)

Defined in Linear.Algebra

Methods

comult :: (E V1 -> r) -> E V1 -> E V1 -> r

counital :: (E V1 -> r) -> r

Num r => Coalgebra r (E V0)

Defined in Linear.Algebra

Methods

comult :: (E V0 -> r) -> E V0 -> E V0 -> r

counital :: (E V0 -> r) -> r

(Num r, TrivialConjugate r) => Algebra r (E Quaternion)

Defined in Linear.Algebra

Methods

mult :: (E Quaternion -> E Quaternion -> r) -> E Quaternion -> r

unital :: r -> E Quaternion -> r

Num r => Algebra r (E Complex)

Defined in Linear.Algebra

Methods

mult :: (E Complex -> E Complex -> r) -> E Complex -> r

unital :: r -> E Complex -> r

Num r => Algebra r (E V1)

Defined in Linear.Algebra

Methods

mult :: (E V1 -> E V1 -> r) -> E V1 -> r

unital :: r -> E V1 -> r

Num r => Algebra r (E V0)

Defined in Linear.Algebra

Methods

mult :: (E V0 -> E V0 -> r) -> E V0 -> r

unital :: r -> E V0 -> r

FunctorWithIndex (E V1) V1

Defined in Linear.V1

Methods

imap :: (E V1 -> a -> b) -> V1 a -> V1 b

FunctorWithIndex (E V2) V2

Defined in Linear.V2

Methods

imap :: (E V2 -> a -> b) -> V2 a -> V2 b

FunctorWithIndex (E V3) V3

Defined in Linear.V3

Methods

imap :: (E V3 -> a -> b) -> V3 a -> V3 b

FunctorWithIndex (E V4) V4

Defined in Linear.V4

Methods

imap :: (E V4 -> a -> b) -> V4 a -> V4 b

FunctorWithIndex (E V0) V0

Defined in Linear.V0

Methods

imap :: (E V0 -> a -> b) -> V0 a -> V0 b

FunctorWithIndex (E Quaternion) Quaternion

Defined in Linear.Quaternion

Methods

imap :: (E Quaternion -> a -> b) -> Quaternion a -> Quaternion b

FunctorWithIndex (E Plucker) Plucker

Defined in Linear.Plucker

Methods

imap :: (E Plucker -> a -> b) -> Plucker a -> Plucker b

FoldableWithIndex (E V1) V1

Defined in Linear.V1

Methods

ifoldMap :: Monoid m => (E V1 -> a -> m) -> V1 a -> m

ifoldMap' :: Monoid m => (E V1 -> a -> m) -> V1 a -> m

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

FoldableWithIndex (E V2) V2

Defined in Linear.V2

Methods

ifoldMap :: Monoid m => (E V2 -> a -> m) -> V2 a -> m

ifoldMap' :: Monoid m => (E V2 -> a -> m) -> V2 a -> m

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

FoldableWithIndex (E V3) V3

Defined in Linear.V3

Methods

ifoldMap :: Monoid m => (E V3 -> a -> m) -> V3 a -> m

ifoldMap' :: Monoid m => (E V3 -> a -> m) -> V3 a -> m

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

FoldableWithIndex (E V4) V4

Defined in Linear.V4

Methods

ifoldMap :: Monoid m => (E V4 -> a -> m) -> V4 a -> m

ifoldMap' :: Monoid m => (E V4 -> a -> m) -> V4 a -> m

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

FoldableWithIndex (E V0) V0

Defined in Linear.V0

Methods

ifoldMap :: Monoid m => (E V0 -> a -> m) -> V0 a -> m

ifoldMap' :: Monoid m => (E V0 -> a -> m) -> V0 a -> m

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

FoldableWithIndex (E Quaternion) Quaternion

Defined in Linear.Quaternion

Methods

ifoldMap :: Monoid m => (E Quaternion -> a -> m) -> Quaternion a -> m

ifoldMap' :: Monoid m => (E Quaternion -> a -> m) -> Quaternion a -> m

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

FoldableWithIndex (E Plucker) Plucker

Defined in Linear.Plucker

Methods

ifoldMap :: Monoid m => (E Plucker -> a -> m) -> Plucker a -> m

ifoldMap' :: Monoid m => (E Plucker -> a -> m) -> Plucker a -> m

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

TraversableWithIndex (E V1) V1

Defined in Linear.V1

Methods

itraverse :: Applicative f => (E V1 -> a -> f b) -> V1 a -> f (V1 b)

TraversableWithIndex (E V2) V2

Defined in Linear.V2

Methods

itraverse :: Applicative f => (E V2 -> a -> f b) -> V2 a -> f (V2 b)

TraversableWithIndex (E V3) V3

Defined in Linear.V3

Methods

itraverse :: Applicative f => (E V3 -> a -> f b) -> V3 a -> f (V3 b)

TraversableWithIndex (E V4) V4

Defined in Linear.V4

Methods

itraverse :: Applicative f => (E V4 -> a -> f b) -> V4 a -> f (V4 b)

TraversableWithIndex (E V0) V0

Defined in Linear.V0

Methods

itraverse :: Applicative f => (E V0 -> a -> f b) -> V0 a -> f (V0 b)

TraversableWithIndex (E Quaternion) Quaternion

Defined in Linear.Quaternion

Methods

itraverse :: Applicative f => (E Quaternion -> a -> f b) -> Quaternion a -> f (Quaternion b)

TraversableWithIndex (E Plucker) Plucker

Defined in Linear.Plucker

Methods

itraverse :: Applicative f => (E Plucker -> a -> f b) -> Plucker a -> f (Plucker b)

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

Compute the negation of a vector

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

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

Compute the right scalar product

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

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

Compute the left scalar product

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

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

Compute division by a scalar on the right.

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

Sum over multiple vectors

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

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

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]

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

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

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)

Outer (tensor) product of two vectors

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

Create a unit vector.

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