vector-space-0.10: Vector & affine spaces, linear maps, and derivatives

Data.Maclaurin

Contents

Description

Infinite derivative towers via linear maps, using the Maclaurin representation. See blog posts http://conal.net/blog/tag/derivatives/.

Synopsis

# Documentation

data a :> b Source

Tower of derivatives.

Constructors

 D infixr 9 FieldspowVal :: b derivative :: a :-* (a :> b)

Instances

 (Num s, VectorSpace s, HasBasis s, HasTrie (Basis s), HasNormal ((:>) (Two s) (Three s))) => HasNormal (Three ((:>) (Two s) s)) (Num s, VectorSpace s, HasBasis s, HasTrie (Basis s), (~) * (Basis s) ()) => HasNormal (Two ((:>) (One s) s)) Eq b => Eq ((:>) a b) (HasBasis a, (~) * s (Scalar a), HasTrie (Basis a), Floating s, VectorSpace s, (~) * (Scalar s) s) => Floating ((:>) a s) (HasBasis a, (~) * s (Scalar a), HasTrie (Basis a), Fractional s, VectorSpace s, (~) * (Scalar s) s) => Fractional ((:>) a s) (HasBasis a, (~) * s (Scalar a), HasTrie (Basis a), Num s, VectorSpace s, (~) * (Scalar s) s) => Num ((:>) a s) (AdditiveGroup b, HasBasis a, HasTrie (Basis a), OrdB b, IfB b, Ord b) => Ord ((:>) a b) Show b => Show ((:>) a b) (AdditiveGroup v, HasBasis u, HasTrie (Basis u), IfB v) => IfB ((:>) u v) (AdditiveGroup v, HasBasis u, HasTrie (Basis u), OrdB v) => OrdB ((:>) u v) (HasBasis a, HasTrie (Basis a), AdditiveGroup u) => AdditiveGroup ((:>) a u) (InnerSpace u, (~) * s (Scalar u), AdditiveGroup s, HasBasis a, HasTrie (Basis a)) => InnerSpace ((:>) a u) (HasBasis a, HasTrie (Basis a), VectorSpace u, AdditiveGroup (Scalar u)) => VectorSpace ((:>) a u) (HasBasis a, HasTrie (Basis a), VectorSpace v, HasCross3 v) => HasCross3 ((:>) a v) (HasBasis a, HasTrie (Basis a), VectorSpace v, HasCross2 v) => HasCross2 ((:>) a v) (Num s, HasTrie (Basis (s, s)), HasBasis s, (~) * (Basis s) ()) => HasNormal ((:>) (Two s) (Three s)) (HasBasis s, HasTrie (Basis s), (~) * (Basis s) ()) => HasNormal ((:>) (One s) (Two s)) type BooleanOf ((:>) a b) = BooleanOf b type Scalar ((:>) a u) = (:>) a (Scalar u)

derivAtBasis :: (HasTrie (Basis a), HasBasis a, AdditiveGroup b) => (a :> b) -> Basis a -> a :> b Source

Sample the derivative at a basis element. Optimized for partial application to save work for non-scalar derivatives.

type (:~>) a b = a -> a :> b Source

Infinitely differentiable functions

pureD :: (AdditiveGroup b, HasBasis a, HasTrie (Basis a)) => b -> a :> b Source

Constant derivative tower.

fmapD :: (HasBasis a, HasTrie (Basis a), AdditiveGroup b) => (b -> c) -> (a :> b) -> a :> c Source

Map a linear function over a derivative tower.

(<\$>>) :: (HasBasis a, HasTrie (Basis a), AdditiveGroup b) => (b -> c) -> (a :> b) -> a :> c infixl 4 Source

Map a linear function over a derivative tower.

liftD2 :: (HasBasis a, HasTrie (Basis a), AdditiveGroup b, AdditiveGroup c) => (b -> c -> d) -> (a :> b) -> (a :> c) -> a :> d Source

Apply a linear binary function over derivative towers.

liftD3 :: (HasBasis a, HasTrie (Basis a), AdditiveGroup b, AdditiveGroup c, AdditiveGroup d) => (b -> c -> d -> e) -> (a :> b) -> (a :> c) -> (a :> d) -> a :> e Source

Apply a linear ternary function over derivative towers.

idD :: (VectorSpace u, s ~ Scalar u, VectorSpace (u :> u), VectorSpace s, HasBasis u, HasTrie (Basis u)) => u :~> u Source

Differentiable identity function. Sometimes called "the derivation variable" or similar, but it's not really a variable.

fstD :: (HasBasis a, HasTrie (Basis a), HasBasis b, HasTrie (Basis b), Scalar a ~ Scalar b) => (a, b) :~> a Source

Differentiable version of `fst`

sndD :: (HasBasis a, HasTrie (Basis a), HasBasis b, HasTrie (Basis b), Scalar a ~ Scalar b) => (a, b) :~> b Source

Differentiable version of `snd`

linearD :: (HasBasis u, HasTrie (Basis u), AdditiveGroup v) => (u -> v) -> u :~> v Source

Every linear function has a constant derivative equal to the function itself (as a linear map).

distrib :: forall a b c u. (HasBasis a, HasTrie (Basis a), AdditiveGroup b, AdditiveGroup c, AdditiveGroup u) => (b -> c -> u) -> (a :> b) -> (a :> c) -> a :> u Source

Derivative tower for applying a binary function that distributes over addition, such as multiplication. A bit weaker assumption than bilinearity. Is bilinearity necessary for correctness here?

(>-<) :: (HasBasis a, HasTrie (Basis a), VectorSpace u, AdditiveGroup (Scalar u)) => (u -> u) -> ((a :> u) -> a :> Scalar u) -> (a :> u) -> a :> u infix 0 Source