linear-1.21.1: Linear Algebra

Linear.Vector

Description

Operations on free vector spaces.

Synopsis

# Documentation

class Functor f => Additive f where Source #

Minimal complete definition

Nothing

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

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 # Instance detailsDefined in Linear.Algebra Methodscomult :: (E Quaternion -> r) -> E Quaternion -> E Quaternion -> r Source #counital :: (E Quaternion -> r) -> r Source # Num r => Coalgebra r (E Complex) Source # Instance detailsDefined in Linear.Algebra Methodscomult :: (E Complex -> r) -> E Complex -> E Complex -> r Source #counital :: (E Complex -> r) -> r Source # Num r => Coalgebra r (E V4) Source # Instance detailsDefined in Linear.Algebra Methodscomult :: (E V4 -> r) -> E V4 -> E V4 -> r Source #counital :: (E V4 -> r) -> r Source # Num r => Coalgebra r (E V3) Source # Instance detailsDefined in Linear.Algebra Methodscomult :: (E V3 -> r) -> E V3 -> E V3 -> r Source #counital :: (E V3 -> r) -> r Source # Num r => Coalgebra r (E V2) Source # Instance detailsDefined in Linear.Algebra Methodscomult :: (E V2 -> r) -> E V2 -> E V2 -> r Source #counital :: (E V2 -> r) -> r Source # Num r => Coalgebra r (E V1) Source # Instance detailsDefined in Linear.Algebra Methodscomult :: (E V1 -> r) -> E V1 -> E V1 -> r Source #counital :: (E V1 -> r) -> r Source # Num r => Coalgebra r (E V0) Source # Instance detailsDefined in Linear.Algebra 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 # Instance detailsDefined in Linear.Algebra Methodsmult :: (E Quaternion -> E Quaternion -> r) -> E Quaternion -> r Source #unital :: r -> E Quaternion -> r Source # Num r => Algebra r (E Complex) Source # Instance detailsDefined in Linear.Algebra Methodsmult :: (E Complex -> E Complex -> r) -> E Complex -> r Source #unital :: r -> E Complex -> r Source # Num r => Algebra r (E V1) Source # Instance detailsDefined in Linear.Algebra Methodsmult :: (E V1 -> E V1 -> r) -> E V1 -> r Source #unital :: r -> E V1 -> r Source # Num r => Algebra r (E V0) Source # Instance detailsDefined in Linear.Algebra Methodsmult :: (E V0 -> E V0 -> r) -> E V0 -> r Source #unital :: r -> E V0 -> r Source # Source # Instance detailsDefined in Linear.V1 Methodsimap :: (E V1 -> a -> b) -> V1 a -> V1 b #imapped :: IndexedSetter (E V1) (V1 a) (V1 b) a b # Source # Instance detailsDefined in Linear.V2 Methodsimap :: (E V2 -> a -> b) -> V2 a -> V2 b #imapped :: IndexedSetter (E V2) (V2 a) (V2 b) a b # Source # Instance detailsDefined in Linear.V3 Methodsimap :: (E V3 -> a -> b) -> V3 a -> V3 b #imapped :: IndexedSetter (E V3) (V3 a) (V3 b) a b # Source # Instance detailsDefined in Linear.V4 Methodsimap :: (E V4 -> a -> b) -> V4 a -> V4 b #imapped :: IndexedSetter (E V4) (V4 a) (V4 b) a b # Source # Instance detailsDefined in Linear.V0 Methodsimap :: (E V0 -> a -> b) -> V0 a -> V0 b #imapped :: IndexedSetter (E V0) (V0 a) (V0 b) a b # Source # Instance detailsDefined in Linear.Quaternion Methodsimap :: (E Quaternion -> a -> b) -> Quaternion a -> Quaternion b #imapped :: IndexedSetter (E Quaternion) (Quaternion a) (Quaternion b) a b # Source # Instance detailsDefined in Linear.Plucker Methodsimap :: (E Plucker -> a -> b) -> Plucker a -> Plucker b #imapped :: IndexedSetter (E Plucker) (Plucker a) (Plucker b) a b # Source # Instance detailsDefined in Linear.V1 MethodsifoldMap :: Monoid m => (E V1 -> a -> m) -> V1 a -> m #ifolded :: IndexedFold (E V1) (V1 a) 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 # Source # Instance detailsDefined in Linear.V2 MethodsifoldMap :: Monoid m => (E V2 -> a -> m) -> V2 a -> m #ifolded :: IndexedFold (E V2) (V2 a) 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 # Source # Instance detailsDefined in Linear.V3 MethodsifoldMap :: Monoid m => (E V3 -> a -> m) -> V3 a -> m #ifolded :: IndexedFold (E V3) (V3 a) 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 # Source # Instance detailsDefined in Linear.V4 MethodsifoldMap :: Monoid m => (E V4 -> a -> m) -> V4 a -> m #ifolded :: IndexedFold (E V4) (V4 a) 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 # Source # Instance detailsDefined in Linear.V0 MethodsifoldMap :: Monoid m => (E V0 -> a -> m) -> V0 a -> m #ifolded :: IndexedFold (E V0) (V0 a) 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 # Source # Instance detailsDefined in Linear.Quaternion MethodsifoldMap :: Monoid m => (E Quaternion -> a -> m) -> Quaternion a -> m #ifolded :: IndexedFold (E Quaternion) (Quaternion a) 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 # Source # Instance detailsDefined in Linear.Plucker MethodsifoldMap :: Monoid m => (E Plucker -> a -> m) -> Plucker a -> m #ifolded :: IndexedFold (E Plucker) (Plucker a) 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 # Source # Instance detailsDefined in Linear.V1 Methodsitraverse :: Applicative f => (E V1 -> a -> f b) -> V1 a -> f (V1 b) #itraversed :: IndexedTraversal (E V1) (V1 a) (V1 b) a b # Source # Instance detailsDefined in Linear.V2 Methodsitraverse :: Applicative f => (E V2 -> a -> f b) -> V2 a -> f (V2 b) #itraversed :: IndexedTraversal (E V2) (V2 a) (V2 b) a b # Source # Instance detailsDefined in Linear.V3 Methodsitraverse :: Applicative f => (E V3 -> a -> f b) -> V3 a -> f (V3 b) #itraversed :: IndexedTraversal (E V3) (V3 a) (V3 b) a b # Source # Instance detailsDefined in Linear.V4 Methodsitraverse :: Applicative f => (E V4 -> a -> f b) -> V4 a -> f (V4 b) #itraversed :: IndexedTraversal (E V4) (V4 a) (V4 b) a b # Source # Instance detailsDefined in Linear.V0 Methodsitraverse :: Applicative f => (E V0 -> a -> f b) -> V0 a -> f (V0 b) #itraversed :: IndexedTraversal (E V0) (V0 a) (V0 b) a b # Source # Instance detailsDefined in Linear.Quaternion Methodsitraverse :: Applicative f => (E Quaternion -> a -> f b) -> Quaternion a -> f (Quaternion b) #itraversed :: IndexedTraversal (E Quaternion) (Quaternion a) (Quaternion b) a b # Source # Instance detailsDefined in Linear.Plucker Methodsitraverse :: Applicative f => (E Plucker -> a -> f b) -> Plucker a -> f (Plucker b) #itraversed :: IndexedTraversal (E Plucker) (Plucker a) (Plucker b) a 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