vector-space-0.19: Vector & affine spaces, linear maps, and derivatives

Copyright | (c) Conal Elliott 2008-2016
License | BSD3
Maintainer | conal@conal.net
Stability | experimental
Safe Haskell | None
Language | Haskell2010

Data.LinearMap

Description

Linear maps

Synopsis

- data u :-* v
- linear :: (HasBasis u, HasTrie (Basis u)) => (u -> v) -> u :-* v
- lapply :: (VectorSpace v, Scalar u ~ Scalar v, HasBasis u, HasTrie (Basis u)) => (u :-* v) -> u -> v
- atBasis :: (AdditiveGroup v, HasTrie (Basis u)) => (u :-* v) -> Basis u -> v
- idL :: (HasBasis u, HasTrie (Basis u)) => u :-* u
- (*.*) :: (HasTrie (Basis u), HasBasis v, HasTrie (Basis v), VectorSpace w, Scalar v ~ Scalar w) => (v :-* w) -> (u :-* v) -> u :-* w
- 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)
- liftL2 :: (Applicative f, AdditiveGroup (f a), AdditiveGroup (f b)) => (a -> b -> c) -> MSum (f a) -> MSum (f b) -> MSum (f c)
- 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)
- 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

# Documentation

data u :-* v infixr 1 Source #

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

#### Instances

Instances details

(HasTrie (Basis u), AdditiveGroup v) => AdditiveGroup (u :-* v) Source # |
Instance details Defined in Data.LinearMap Methods zeroV :: u :-* v Source # (^+^) :: (u :-* v) -> (u :-* v) -> u :-* v Source # negateV :: (u :-* v) -> u :-* v Source # (^-^) :: (u :-* v) -> (u :-* v) -> u :-* v Source #
(HasTrie (Basis u), VectorSpace v) => VectorSpace (u :-* v) Source # |

Instance details

Defined in Data.LinearMap

Associated Types

type Scalar (u :-* v) |
Instance details Defined in Data.LinearMap type Scalar (u :-* v) = Scalar v

Methods

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

type Scalar (u :-* v) Source # |
Instance details Defined in Data.LinearMap type Scalar (u :-* v) = Scalar v

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

Function (assumed linear) as linear map.

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

Apply a linear map to a vector.

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

Evaluate a linear map on a basis element.

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

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 Source #

Compose linear maps

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

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

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

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

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

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

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

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) Source #

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) Source #

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 Source #

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

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

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) Source #

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') Source #

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

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

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 Source #