vector-space-0.7.1: Vector & affine spaces, linear maps, and derivatives (requires ghc 6.9 or better)Source codeContentsIndex
Data.Maclaurin
Stabilityexperimental
Maintainerconal@conal.net
Contents
Misc
Description
Infinite derivative towers via linear maps, using the Maclaurin representation. See blog posts http://conal.net/blog/tag/derivatives/.
Synopsis
data a :> b = D {
powVal :: b
derivative :: a :-* (a :> b)
}
derivAtBasis :: (HasTrie (Basis a), HasBasis a, AdditiveGroup b) => (a :> b) -> Basis a -> a :> b
type :~> a b = a -> a :> b
pureD :: (AdditiveGroup b, HasBasis a, HasTrie (Basis a)) => b -> a :> b
fmapD :: (HasBasis a, HasTrie (Basis a), AdditiveGroup b) => (b -> c) -> (a :> b) -> a :> c
(<$>>) :: (HasBasis a, HasTrie (Basis a), AdditiveGroup b) => (b -> c) -> (a :> b) -> a :> c
liftD2 :: (HasBasis a, HasTrie (Basis a), AdditiveGroup b, AdditiveGroup c) => (b -> c -> d) -> (a :> b) -> (a :> c) -> a :> d
liftD3 :: (HasBasis a, HasTrie (Basis a), AdditiveGroup b, AdditiveGroup c, AdditiveGroup d) => (b -> c -> d -> e) -> (a :> b) -> (a :> c) -> (a :> d) -> a :> e
idD :: (VectorSpace u, s ~ Scalar u, VectorSpace (u :> u), VectorSpace s, HasBasis u, HasTrie (Basis u)) => u :~> u
fstD :: (HasBasis a, HasTrie (Basis a), HasBasis b, HasTrie (Basis b), Scalar a ~ Scalar b) => (a, b) :~> a
sndD :: (HasBasis a, HasTrie (Basis a), HasBasis b, HasTrie (Basis b), Scalar a ~ Scalar b) => (a, b) :~> b
linearD :: (HasBasis u, HasTrie (Basis u), AdditiveGroup v) => (u -> v) -> u :~> v
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
(>-<) :: (HasBasis a, HasTrie (Basis a), VectorSpace u, AdditiveGroup (Scalar u)) => (u -> u) -> ((a :> u) -> a :> Scalar u) -> (a :> u) -> a :> u
pairD :: (HasBasis a, HasTrie (Basis a), VectorSpace b, VectorSpace c, Scalar b ~ Scalar c) => (a :> b, a :> c) -> a :> (b, c)
unpairD :: (HasBasis a, HasTrie (Basis a), VectorSpace a, VectorSpace b, VectorSpace c, Scalar b ~ Scalar c) => (a :> (b, c)) -> (a :> b, a :> c)
tripleD :: (HasBasis a, HasTrie (Basis a), VectorSpace b, VectorSpace c, VectorSpace d, Scalar b ~ Scalar c, Scalar c ~ Scalar d) => (a :> b, a :> c, a :> d) -> a :> (b, c, d)
untripleD :: (HasBasis a, HasTrie (Basis a), VectorSpace a, VectorSpace b, VectorSpace c, VectorSpace d, Scalar b ~ Scalar c, Scalar c ~ Scalar d) => (a :> (b, c, d)) -> (a :> b, a :> c, a :> d)
Documentation
data a :> b Source
Tower of derivatives.
Constructors
D
powVal :: b
derivative :: a :-* (a :> b)
show/hide Instances
(AdditiveGroup v, HasBasis u, HasTrie (Basis u), IfB b v) => IfB b (u :> v)
(AdditiveGroup v, HasBasis u, HasTrie (Basis u), OrdB b v) => OrdB b (u :> v)
Eq b => Eq (a :> b)
(s ~ Scalar a, Scalar s ~ s, HasBasis a, HasTrie (Basis a), Floating s, VectorSpace s) => Floating (a :> s)
(s ~ Scalar a, Scalar s ~ s, HasBasis a, HasTrie (Basis a), Fractional s, VectorSpace s) => Fractional (a :> s)
(s ~ Scalar a, Scalar s ~ s, HasBasis a, HasTrie (Basis a), Num s, VectorSpace s) => Num (a :> s)
(AdditiveGroup b, HasBasis a, HasTrie (Basis a), OrdB bool b, IfB bool b, Ord b) => Ord (a :> b)
Show b => Show (a :> b)
(HasBasis a, HasTrie (Basis a), AdditiveGroup u) => AdditiveGroup (a :> u)
(s ~ Scalar u, InnerSpace 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)
(Basis s ~ (), Num s, HasTrie (Basis ((,) s s)), HasBasis s) => HasNormal (Two s :> Three s)
(Basis s ~ (), HasBasis s, HasTrie (Basis s)) => HasNormal (One s :> Two s)
derivAtBasis :: (HasTrie (Basis a), HasBasis a, AdditiveGroup b) => (a :> b) -> Basis a -> a :> bSource
Sample the derivative at a basis element. Optimized for partial application to save work for non-scalar derivatives.
type :~> a b = a -> a :> bSource
Infinitely differentiable functions
pureD :: (AdditiveGroup b, HasBasis a, HasTrie (Basis a)) => b -> a :> bSource
Constant derivative tower.
fmapD :: (HasBasis a, HasTrie (Basis a), AdditiveGroup b) => (b -> c) -> (a :> b) -> a :> cSource
(<$>>) :: (HasBasis a, HasTrie (Basis a), AdditiveGroup b) => (b -> c) -> (a :> b) -> a :> cSource
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 :> dSource
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 :> eSource
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 :~> uSource
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) :~> aSource
Differentiable version of fst
sndD :: (HasBasis a, HasTrie (Basis a), HasBasis b, HasTrie (Basis b), Scalar a ~ Scalar b) => (a, b) :~> bSource
Differentiable version of snd
linearD :: (HasBasis u, HasTrie (Basis u), AdditiveGroup v) => (u -> v) -> u :~> vSource
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 :> uSource
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 :> uSource
Specialized chain rule. See also '(@.)'
Misc
pairD :: (HasBasis a, HasTrie (Basis a), VectorSpace b, VectorSpace c, Scalar b ~ Scalar c) => (a :> b, a :> c) -> a :> (b, c)Source
unpairD :: (HasBasis a, HasTrie (Basis a), VectorSpace a, VectorSpace b, VectorSpace c, Scalar b ~ Scalar c) => (a :> (b, c)) -> (a :> b, a :> c)Source
tripleD :: (HasBasis a, HasTrie (Basis a), VectorSpace b, VectorSpace c, VectorSpace d, Scalar b ~ Scalar c, Scalar c ~ Scalar d) => (a :> b, a :> c, a :> d) -> a :> (b, c, d)Source
untripleD :: (HasBasis a, HasTrie (Basis a), VectorSpace a, VectorSpace b, VectorSpace c, VectorSpace d, Scalar b ~ Scalar c, Scalar c ~ Scalar d) => (a :> (b, c, d)) -> (a :> b, a :> c, a :> d)Source
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