Numeric.Covector

Documentation

newtype Covector r a

Linear functionals from elements of an (infinite) free module to a scalar

Constructors

Covector ((a -> r) -> r)

Instances

(Semiring r, Additive (Covector s m), RightModule r s) => RightModule r (Covector s m)

(Semiring r, Additive (Covector s m), LeftModule r s) => LeftModule r (Covector s m)

Monad (Covector r)

Functor (Covector r)

(Monad (Covector r), Monoidal r) => MonadPlus (Covector r)

Functor (Covector r) => Applicative (Covector r)

(Applicative (Covector r), Monoidal r) => Alternative (Covector r)

(Alt (Covector r), Monoidal r) => Plus (Covector r)

(Functor (Covector r), Additive r) => Alt (Covector r)

Functor (Covector r) => Apply (Covector r)

Apply (Covector r) => Bind (Covector r)

(Additive (Covector r a), Idempotent r) => Idempotent (Covector r a)

(Additive (Covector s a), Abelian s) => Abelian (Covector s a)

Additive r => Additive (Covector r a)

(LeftModule Natural (Covector s a), RightModule Natural (Covector s a), Monoidal s) => Monoidal (Covector s a)

(Additive (Covector r m), Abelian (Covector r m), Multiplicative (Covector r m), Coalgebra r m) => Semiring (Covector r m)

Coalgebra r m => Multiplicative (Covector r m)

(LeftModule Integer (Covector s a), RightModule Integer (Covector s a), Monoidal (Covector s a), Group s) => Group (Covector s a)

(Multiplicative (Covector r m), CounitalCoalgebra r m) => Unital (Covector r m)

(Multiplicative (Covector r a), Idempotent r, IdempotentCoalgebra r a) => Band (Covector r a)

(Semiring (Covector r m), Unital (Covector r m), Monoidal (Covector r m), Rig r, CounitalCoalgebra r m) => Rig (Covector r m)

(Rig (Covector r m), Rng (Covector r m), Ring r, CounitalCoalgebra r m) => Ring (Covector r m)

(Multiplicative (Covector r m), Commutative m, Coalgebra r m) => Commutative (Covector r m)

Distinguished a => Distinguished (Covector r a)

(Distinguished (Covector r a), Complicated a) => Complicated (Covector r a)

(Complicated (Covector r a), Hamiltonian a) => Hamiltonian (Covector r a)

(Distinguished (Covector r a), Infinitesimal a) => Infinitesimal (Covector r a)

Hyperbolic a => Hyperbolic (Covector r a)

Trigonometric a => Trigonometric (Covector r a)

(Semiring (Covector r m), Additive (Covector r m), Coalgebra r m) => RightModule (Covector r m) (Covector r m)

(Semiring (Covector r m), Additive (Covector r m), Coalgebra r m) => LeftModule (Covector r m) (Covector r m)

($*) :: Indexable m => Covector r (Key m) -> m r -> r

Covectors as linear functionals

counitM :: UnitalAlgebra r a => a -> Covector r ()

unitM :: CounitalCoalgebra r c => Covector r c

comultM :: Algebra r a => a -> Covector r (a, a)

multM :: Coalgebra r c => c -> c -> Covector r c

invM :: InvolutiveAlgebra r h => h -> Covector r h

coinvM :: InvolutiveCoalgebra r h => h -> Covector r h

antipodeM :: HopfAlgebra r h => h -> Covector r h

convolveM antipodeM return = convolveM return antipodeM = comultM >=> uncurry joinM

convolveM :: (Algebra r c, Coalgebra r a) => (c -> Covector r a) -> (c -> Covector r a) -> c -> Covector r a

memoM :: HasTrie a => a -> Covector s a