algebra-4.2: Constructive abstract algebra

Numeric.Map

Synopsis

# Documentation

newtype Map r b a Source

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 reversed direction of the arrow, 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 ((a -> r) -> b -> r)

Instances

 Category * (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) Monoidal r => ArrowZero (Map r) Monoidal r => ArrowPlus (Map r) ArrowChoice (Map r) ArrowApply (Map r) Semigroupoid (Map r) Monoidal r => Alternative (Map r b) Monad (Map r b) Functor (Map r b) Monoidal r => MonadPlus (Map r b) Applicative (Map r b) Monoidal r => Plus (Map r b) Additive r => Alt (Map r b) Apply (Map r b) Bind (Map r b) Abelian s => Abelian (Map s b a) Additive r => Additive (Map r b a) Monoidal s => Monoidal (Map s b a) Coalgebra r m => Semiring (Map r b m) Coalgebra r m => Multiplicative (Map r b m) Group s => Group (Map s b a) CounitalCoalgebra r m => Unital (Map r b m) (Rig r, CounitalCoalgebra r m) => Rig (Map r b m) (Ring r, CounitalCoalgebra r m) => Ring (Map r a m) (Commutative m, Coalgebra r m) => Commutative (Map r b m) Coalgebra r m => RightModule (Map r b m) (Map r b m) Coalgebra r m => LeftModule (Map r b m) (Map r b m)

(\$@) :: Map r b a -> b -> Covector r a infixr 0 Source

extract a linear functional from a linear map

multMap :: Coalgebra r c => Map r (c, c) c Source

comultMap :: Algebra r a => Map r a (a, a) Source

(inefficiently) combine a linear combination of basis vectors to make a map. arrMap :: (Monoidal r, Semiring r) => (b -> [(r, a)]) -> Map r b a arrMap f = Map \$ k b -> sum [ r * k a | (r, a) <- f b ]

convolveMap :: (Algebra r a, Coalgebra r c) => Map r a c -> Map r a c -> Map r a c Source

convolution given an associative algebra and coassociative coalgebra