Data.LinearMap

Linear maps

Documentation

data u :-* v infixr 1

Linear map, represented as an optional memo-trie from basis to values, where Nothing means the zero map (an optimization).

Instances

(HasTrie (Basis u), AdditiveGroup v) => AdditiveGroup ((:-*) u v)

Methods

zeroV :: u :-* v

(^+^) :: (u :-* v) -> (u :-* v) -> u :-* v

negateV :: (u :-* v) -> u :-* v

(^-^) :: (u :-* v) -> (u :-* v) -> u :-* v

(HasTrie (Basis u), VectorSpace v) => VectorSpace ((:-*) u v)

Associated Types

type Scalar ((:-*) u v) :: *

Methods

(*^) :: Scalar (u :-* v) -> (u :-* v) -> u :-* v

type Scalar ((:-*) u v)

type Scalar ((:-*) u v) = Scalar v

linear :: (HasBasis u, HasTrie (Basis u)) => (u -> v) -> u :-* v

Function (assumed linear) as linear map.

lapply :: (VectorSpace v, Scalar u ~ Scalar v, HasBasis u, HasTrie (Basis u)) => (u :-* v) -> u -> v

Apply a linear map to a vector.

atBasis :: (AdditiveGroup v, HasTrie (Basis u)) => (u :-* v) -> Basis u -> v

Evaluate a linear map on a basis element.

idL :: (HasBasis u, HasTrie (Basis u)) => u :-* u

Identity linear map

(*.*) :: (HasTrie (Basis u), HasBasis v, HasTrie (Basis v), VectorSpace w, Scalar v ~ Scalar w) => (v :-* w) -> (u :-* v) -> u :-* w infixr 9

Compose linear maps

inLMap :: (LMap' r s -> LMap' t u) -> (r :-* s) -> t :-* u

inLMap2 :: (LMap' r s -> LMap' t u -> LMap' v w) -> (r :-* s) -> (t :-* u) -> v :-* w

inLMap3 :: (LMap' r s -> LMap' t u -> LMap' v w -> LMap' x y) -> (r :-* s) -> (t :-* u) -> (v :-* w) -> x :-* y

liftMS :: (a -> b) -> MSum a -> MSum b

liftMS2 :: (AdditiveGroup a, AdditiveGroup b) => (a -> b -> c) -> MSum a -> MSum b -> MSum c

liftMS3 :: (AdditiveGroup a, AdditiveGroup b, AdditiveGroup c) => (a -> b -> c -> d) -> MSum a -> MSum b -> MSum c -> MSum d

liftL :: Functor f => (a -> b) -> MSum (f a) -> MSum (f b)

Apply a linear function to each element of a linear map. liftL f l == linear f *.* l, but works more efficiently.

liftL2 :: (Applicative f, AdditiveGroup (f a), AdditiveGroup (f b)) => (a -> b -> c) -> MSum (f a) -> MSum (f b) -> MSum (f c)

Apply a linear binary function (not to be confused with a bilinear function) to each element of a linear map.

liftL3 :: (Applicative f, AdditiveGroup (f a), AdditiveGroup (f b), AdditiveGroup (f c)) => (a -> b -> c -> d) -> MSum (f a) -> MSum (f b) -> MSum (f c) -> MSum (f d)

Apply a linear ternary function (not to be confused with a trilinear function) to each element of a linear map.

exlL :: (HasBasis a, HasTrie (Basis a), HasBasis b, HasTrie (Basis b), Scalar a ~ Scalar b) => (a, b) :-* a

exrL :: (HasBasis a, HasTrie (Basis a), HasBasis b, HasTrie (Basis b), Scalar a ~ Scalar b) => (a, b) :-* b

forkL :: (HasTrie (Basis a), HasBasis c, HasBasis d) => (a :-* c) -> (a :-* d) -> a :-* (c, d)

firstL :: (HasBasis u, HasBasis u', HasBasis v, HasTrie (Basis u), HasTrie (Basis v), Scalar u ~ Scalar v, Scalar u ~ Scalar u') => (u :-* u') -> (u, v) :-* (u', v)

secondL :: (HasBasis u, HasBasis v, HasBasis v', HasTrie (Basis u), HasTrie (Basis v), Scalar u ~ Scalar v, Scalar u ~ Scalar v') => (v :-* v') -> (u, v) :-* (u, v')

inlL :: (HasBasis a, HasTrie (Basis a), HasBasis b) => a :-* (a, b)

inrL :: (HasBasis a, HasBasis b, HasTrie (Basis b)) => b :-* (a, b)

joinL :: (HasBasis a, HasTrie (Basis a), HasBasis b, HasTrie (Basis b), Scalar a ~ Scalar b, Scalar a ~ Scalar c, VectorSpace c) => (a :-* c) -> (b :-* c) -> (a, b) :-* c