Numeric.Functional.Linear

Documentation

newtype Linear r a

Linear functionals from elements of a free module to a scalar

Constructors

Linear

Fields

($*) :: (a -> r) -> r

Instances

RightModule r s => RightModule r (Linear s m)

LeftModule r s => LeftModule r (Linear s m)

Monad (Linear r)

Functor (Linear r)

AdditiveMonoid r => MonadPlus (Linear r)

Applicative (Linear r)

AdditiveMonoid r => Alternative (Linear r)

AdditiveMonoid r => Plus (Linear r)

Additive r => Alt (Linear r)

Apply (Linear r)

Bind (Linear r)

Additive r => Additive (Linear r a)

Abelian s => Abelian (Linear s a)

FreeCoalgebra r m => Semiring (Linear r m)

FreeCoalgebra r m => Multiplicative (Linear r m)

FreeCounitalCoalgebra r m => Unital (Linear r m)

(Commutative m, FreeCoalgebra r m) => Commutative (Linear r m)

AdditiveMonoid s => AdditiveMonoid (Linear s a)

(Rig r, FreeCounitalCoalgebra r m) => Rig (Linear r m)

AdditiveGroup s => AdditiveGroup (Linear s a)

(Rng r, FreeCounitalCoalgebra r m) => Rng (Linear r m)

(Ring r, FreeCounitalCoalgebra r m) => Ring (Linear r m)

FreeCoalgebra r m => RightModule (Linear r m) (Linear r m)

FreeCoalgebra r m => LeftModule (Linear r m) (Linear r m)

(.*) :: LeftModule r m => r -> m -> m

(*.) :: RightModule r m => m -> r -> m

Vectors

type Vector = (->)

unitVector :: FreeUnitalAlgebra r a => a -> r

Covectors as linear functionals

type Covector a r = Linear r a

counitCovector :: FreeCounitalCoalgebra r c => Linear r c

embedCovector :: (Unital m, FreeCounitalCoalgebra r m) => r -> Linear r m

augmentCovector :: Unital s => Linear s a -> s

The augmentation ring homomorphism from r^a -> r, generalizes the augmentation homomorphism from a monoid semiring to the underlying semiring