Numeric.Map.Linear

Documentation

newtype Map r b a

linear maps from elements of a free module to another free module over r

 f $# x + y = (f $# x) + (f $# y)
 f $# (r .* x) = r .* (f $# x)

Map r b a represents a linear mapping from a free module with basis a over r to a free module with basis b over r.

Note well the change of direction, due to the contravariance of change of basis!

This way enables we can employ arbitrary pure functions as linear maps by lifting them using arr, or build them by using the monad instance for Map r b. As a consequence Map is an instance of, well, almost everything.

Constructors

Map

Fields

($#) :: (a -> r) -> b -> r

Instances

PFunctor Either (Map r) (Map r)

PFunctor (,) (Map r) (Map r)

QFunctor Either (Map r) (Map r)

QFunctor (,) (Map r) (Map r)

Bifunctor Either (Map r) (Map r) (Map r)

Bifunctor (,) (Map r) (Map r) (Map r)

MonadReader b (Map r b)

RightModule r s => RightModule r (Map s b m)

LeftModule r s => LeftModule r (Map s b m)

Arrow (Map r)

AdditiveMonoid r => ArrowZero (Map r)

AdditiveMonoid r => ArrowPlus (Map r)

ArrowChoice (Map r)

ArrowApply (Map r)

Category (Map r)

Distributive (Map r)

CCC (Map r)

PreCartesian (Map r)

PreCoCartesian (Map r)

Semigroupoid (Map r)

Braided (Map r) Either

Braided (Map r) (,)

Symmetric (Map r) Either

Symmetric (Map r) (,)

Monoidal (Map r) Either

Monoidal (Map r) (,)

Comonoidal (Map r) Either

Comonoidal (Map r) (,)

Associative (Map r) Either

Associative (Map r) (,)

Disassociative (Map r) Either

Disassociative (Map r) (,)

Monad (Map r b)

Functor (Map r b)

AdditiveMonoid r => MonadPlus (Map r b)

Applicative (Map r b)

AdditiveMonoid r => Alternative (Map r b)

AdditiveMonoid r => Plus (Map r b)

Additive r => Alt (Map r b)

Apply (Map r b)

Bind (Map r b)

Additive r => Additive (Map r b a)

Abelian s => Abelian (Map s b a)

FreeCoalgebra r m => Semiring (Map r b m)

FreeCoalgebra r m => Multiplicative (Map r b m)

(Commutative m, FreeCoalgebra r m) => Commutative (Map r b m)

AdditiveMonoid s => AdditiveMonoid (Map s b a)

FreeCounitalCoalgebra r m => Unital (Map r b m)

(Rig r, FreeCounitalCoalgebra r m) => Rig (Map r b m)

AdditiveGroup s => AdditiveGroup (Map s b a)

(Rng r, FreeCounitalCoalgebra r m) => Rng (Map r b m)

(Ring r, FreeCounitalCoalgebra r m) => Ring (Map r a m)

FreeCoalgebra r m => RightModule (Map r b m) (Map r b m)

FreeCoalgebra r m => LeftModule (Map r b m) (Map r b m)

($@) :: Map r b a -> b -> Linear r a

extract a linear functional from a linear map

joinMap :: FreeAlgebra r a => Map r a (a, a)

unitMap :: FreeUnitalAlgebra r a => Map r a ()

memoMap :: HasTrie a => Map r a a

Memoize the results of this linear map

cojoinMap :: FreeCoalgebra r c => Map r (c, c) c

counitMap :: FreeCounitalCoalgebra r c => Map r () c

antipodeMap :: Hopf r h => Map r h h

convolveMap :: (FreeAlgebra r a, FreeCoalgebra r c) => Map r a c -> Map r a c -> Map r a c

convolution given an associative algebra and coassociative coalgebra

embedMap :: (Unital m, FreeCounitalCoalgebra r m) => (b -> r) -> Map r b m

augmentMap :: Unital s => Map s b m -> b -> s

The augmentation ring homomorphism from r^a -> r

arrMap :: (AdditiveMonoid r, Semiring r) => (b -> [(r, a)]) -> Map r b a

(inefficiently) combine a linear combination of basis vectors to make a map.