Copyright	(c) Conal Elliott 2008
License	BSD3
Maintainer	conal@conal.net
Stability	experimental
Safe Haskell	None
Language	Haskell98

Data.Maclaurin

Contents

Misc

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

Tower of derivatives.

Constructors

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

Instances

(VectorSpace s, HasBasis s, HasTrie (Basis s), HasNormal ((:>) (Two s) (Three s))) => HasNormal (Three ((:>) (Two s) s))
(VectorSpace s, HasBasis s, HasTrie (Basis s), (~) * (Basis s) ()) => HasNormal (Two ((:>) (One s) s))
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)
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) => VectorSpace ((:>) a u)
(HasBasis a, HasTrie (Basis a), VectorSpace v, HasCross3 v) => HasCross3 ((:>) a v)
(HasTrie (Basis a), 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

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

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

Infinitely differentiable functions

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

Constant derivative tower.

fmapD :: HasTrie (Basis a) => (b -> c) -> (a :> b) -> a :> c

Map a linear function over a derivative tower.

(<$>>) :: HasTrie (Basis a) => (b -> c) -> (a :> b) -> a :> c infixl 4

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

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

Apply a linear ternary function over derivative towers.

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

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

Differentiable version of fst

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

Differentiable version of snd

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

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 u) => (b -> c -> u) -> (a :> b) -> (a :> c) -> a :> u

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) => (u -> u) -> ((a :> u) -> a :> Scalar u) -> (a :> u) -> a :> u infix 0

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

Misc

pairD :: (HasBasis a, HasTrie (Basis a), VectorSpace b, VectorSpace c) => (a :> b, a :> c) -> a :> (b, c)

unpairD :: HasTrie (Basis a) => (a :> (b, c)) -> (a :> b, a :> c)

tripleD :: (HasBasis a, HasTrie (Basis a), VectorSpace b, VectorSpace c, VectorSpace d) => (a :> b, a :> c, a :> d) -> a :> (b, c, d)

untripleD :: HasTrie (Basis a) => (a :> (b, c, d)) -> (a :> b, a :> c, a :> d)