vector-space-0.2.0: Vector & affine spaces, plus derivatives Source code Contents Index

Data.Maclaurin Stability experimental Maintainer conal@conal.net

Description

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

Synopsis

data a :> b powVal :: :> a b -> b derivative :: :> a b -> a :-* (a :> b) derivativeAt :: (VectorSpace b s, LMapDom a s) => (a :> b) -> a -> a :> b type :~> a b = a -> a :> b dZero :: (LMapDom a s, AdditiveGroup b) => a :> b pureD :: (LMapDom a s, AdditiveGroup b) => b -> a :> b (<$>>) :: (LMapDom a s, VectorSpace b s) => (b -> c) -> (a :> b) -> a :> c liftD2 :: (VectorSpace b s, LMapDom a s, VectorSpace c s, VectorSpace d s) => (b -> c -> d) -> (a :> b) -> (a :> c) -> a :> d liftD3 :: (LMapDom a s, VectorSpace b s, VectorSpace c s, VectorSpace d s, VectorSpace e s) => (b -> c -> d -> e) -> (a :> b) -> (a :> c) -> (a :> d) -> a :> e idD :: (LMapDom u s, VectorSpace u s) => u :~> u fstD :: (VectorSpace a s, LMapDom b s, LMapDom a s) => (a, b) :~> a sndD :: (VectorSpace b s, LMapDom b s, LMapDom a s) => (a, b) :~> b linearD :: (LMapDom u s, VectorSpace v s) => (u -> v) -> u :~> v distrib :: (LMapDom a s, VectorSpace b s, VectorSpace c s, VectorSpace u s) => (b -> c -> u) -> (a :> b) -> (a :> c) -> a :> u (@.) :: (LMapDom b s, LMapDom a s, VectorSpace c s) => (b :~> c) -> (a :~> b) -> a :~> c (>-<) :: (LMapDom a s, VectorSpace s s, VectorSpace u s) => (u -> u) -> ((a :> u) -> a :> s) -> (a :> u) -> a :> u (**^) :: (VectorSpace c s, VectorSpace s s, LMapDom a s) => (a :> s) -> (a :> c) -> a :> c (<*.>) :: (LMapDom a s, InnerSpace b s, VectorSpace s s) => (a :> b) -> (a :> b) -> a :> s

Documentation

data a :> b Source

Tower of derivatives. Instances Eq b => Eq (a :> b) (LMapDom a b, Floating b, VectorSpace b b) => Floating (a :> b) (LMapDom a b, Fractional b, VectorSpace b b) => Fractional (a :> b) (LMapDom a b, Num b, VectorSpace b b) => Num (a :> b) Ord b => Ord (a :> b) Show b => Show (a :> b) (LMapDom a s, VectorSpace u s) => AdditiveGroup (a :> u) (LMapDom a s, VectorSpace u s) => VectorSpace (a :> u) s

powVal :: :> a b -> b Source

derivative :: :> a b -> a :-* (a :> b) Source

derivativeAt :: (VectorSpace b s, LMapDom a s) => (a :> b) -> a -> a :> b Source

Sampled derivative. For avoiding an awkward typing problem related to the two required VectorSpace instances.

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

Infinitely differentiable functions

dZero :: (LMapDom a s, AdditiveGroup b) => a :> b Source

Derivative tower full of zeroV.

pureD :: (LMapDom a s, AdditiveGroup b) => b -> a :> b Source

Constant derivative tower.

(<$>>) :: (LMapDom a s, VectorSpace b s) => (b -> c) -> (a :> b) -> a :> c Source

Map a linear function over a derivative tower.

liftD2 :: (VectorSpace b s, LMapDom a s, VectorSpace c s, VectorSpace d s) => (b -> c -> d) -> (a :> b) -> (a :> c) -> a :> d Source

Apply a linear binary function over derivative towers.

liftD3 :: (LMapDom a s, VectorSpace b s, VectorSpace c s, VectorSpace d s, VectorSpace e s) => (b -> c -> d -> e) -> (a :> b) -> (a :> c) -> (a :> d) -> a :> e Source

Apply a linear ternary function over derivative towers.

idD :: (LMapDom u s, VectorSpace u s) => u :~> u Source

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

fstD :: (VectorSpace a s, LMapDom b s, LMapDom a s) => (a, b) :~> a Source

Differentiable version of fst

sndD :: (VectorSpace b s, LMapDom b s, LMapDom a s) => (a, b) :~> b Source

Differentiable version of snd

linearD :: (LMapDom u s, VectorSpace v s) => (u -> v) -> u :~> v Source

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

distrib :: (LMapDom a s, VectorSpace b s, VectorSpace c s, VectorSpace u s) => (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.

(@.) :: (LMapDom b s, LMapDom a s, VectorSpace c s) => (b :~> c) -> (a :~> b) -> a :~> c Source

Chain rule. See also '(>-<)'.

(>-<) :: (LMapDom a s, VectorSpace s s, VectorSpace u s) => (u -> u) -> ((a :> u) -> a :> s) -> (a :> u) -> a :> u Source

Specialized chain rule. See also '(@.)'

(**^) :: (VectorSpace c s, VectorSpace s s, LMapDom a s) => (a :> s) -> (a :> c) -> a :> c Source

(<*.>) :: (LMapDom a s, InnerSpace b s, VectorSpace s s) => (a :> b) -> (a :> b) -> a :> s Source

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