SubHask.Category.Trans.Derivative

This module provides a category transformer for automatic differentiation.

There are many alternative notions of a generalized derivative. Perhaps the most common is the differential Ring. In Haskell, this might be defined as:

class Field r => Differential r where
   derivative :: r -> r

type Diff cat = forall a b. (Category cat, Differential cat a b)

But this runs into problems with the lack of polymorphic constraints in GHC. See, for example GHC ticket #2893.

References:

• wikipedia article on differntial algebras

Documentation

data Forward a

This is essentially just a translation of the Numeric.AD.Forward.Forward type for use with the SubHask numeric hierarchy.

FIXME:

Add reverse mode auto-differentiation for vectors. Apply the ProofOf framework from Monotonic

Constructors

Forward

Fields

val :: !a

val' :: a

Instances

Show a => Show (Forward a)

IsMutable (Forward a)

Field a => Field (Forward a)

Ring a => Ring (Forward a)

Rig a => Rig (Forward a)

Rg a => Rg (Forward a)

Abelian a => Abelian (Forward a)

Group a => Group (Forward a)

Cancellative a => Cancellative (Forward a)

Monoid a => Monoid (Forward a)

Semigroup a => Semigroup (Forward a)

data Mutable m (Forward a0) = Mutable_AppT__ConT_SubHask_Category_Trans_Derivative_Forward___VarT_a_1628012176_ (PrimRef m (Forward a))

proveC1 :: (a ~ (a >< a), Rig a) => (Forward a -> Forward a) -> C1 (a -> a)

proveC2 :: (a ~ (a >< a), Rig a) => (Forward (Forward a) -> Forward (Forward a)) -> C2 (a -> a)

class C cat where

Associated Types

type D cat :: * -> * -> *

Methods

derivative :: cat a b -> D cat a (a >< b)

Instances

(<=) 1 n => C (Diff n)

data Diff n a b where

Constructors

Diff0 :: (a -> b) -> Diff 0 a b

Diffn :: (a -> b) -> Diff (n - 1) a (a >< b) -> Diff n a b

Instances

(<=) 1 n => C (Diff n)

Sup (* -> * -> *) (->) (Diff n) (->)

(<:) (* -> * -> *) (Diff n) (->)

(<:) (* -> * -> *) (Diff 0) (->)

Sup (* -> * -> *) (Diff n) (->) (->)

(<:) (* -> * -> *) (Diff 1) (Diff 0)

(<:) (* -> * -> *) (Diff 2) (Diff 0)

(<:) (* -> * -> *) (Diff 2) (Diff 1)

Sup (* -> * -> *) (Diff 0) (Diff 1) (Diff 0)

Sup (* -> * -> *) (Diff 0) (Diff 2) (Diff 0)

Sup (* -> * -> *) (Diff 1) (Diff 0) (Diff 0)

Sup (* -> * -> *) (Diff 1) (Diff 2) (Diff 1)

Sup (* -> * -> *) (Diff 2) (Diff 0) (Diff 0)

Sup (* -> * -> *) (Diff 2) (Diff 1) (Diff 1)

IsMutable (Diff n a b)

Monoid b => Monoid (Diff 0 a b)

(Monoid b, Monoid ((><) * * a b)) => Monoid (Diff 1 a b)

(Monoid b, Monoid ((><) * * a b), Monoid ((><) * * a ((><) * * a b))) => Monoid (Diff 2 a b)

Semigroup b => Semigroup (Diff 0 a b)

(Semigroup b, Semigroup ((><) * * a b)) => Semigroup (Diff 1 a b)

(Semigroup b, Semigroup ((><) * * a b), Semigroup ((><) * * a ((><) * * a b))) => Semigroup (Diff 2 a b)

data Mutable m (Diff n0 a0 b0) = Mutable_AppT__AppT__AppT__ConT_SubHask_Category_Trans_Derivative_Diff___VarT_n_1628013694____VarT_a_1628013695____VarT_b_1628013696_ (PrimRef m (Diff n a b))

type D (Diff n) = Diff ((-) n 1)

unsafeProveC0 :: (a -> b) -> Diff 0 a b

unsafeProveC1

Arguments

:: (a -> b)

f(x)

-> (a -> a >< b)

f'(x)

-> C1 (a -> b)

unsafeProveC2

Arguments

:: (a -> b)

f(x)

-> (a -> a >< b)

f'(x)

-> (a -> a >< (a >< b))

f''(x)

-> C2 (a -> b)

type C0 a = C0_ a

type family C0_ f :: *

Equations

C0_ (a -> b) = Diff 0 a b

type C1 a = C1_ a

type family C1_ f :: *

Equations

C1_ (a -> b) = Diff 1 a b

type C2 a = C2_ a

type family C2_ f :: *

Equations

C2_ (a -> b) = Diff 2 a b