Numeric.AD.Tower

Higher order derivatives via a "dual number tower".

Taylor Series

taylor :: Fractional a => UU a -> a -> a -> [a]

taylor0 :: Fractional a => UU a -> a -> a -> [a]

Maclaurin Series

maclaurin :: Fractional a => UU a -> a -> [a]

maclaurin0 :: Fractional a => UU a -> a -> [a]

Derivatives

diff :: Num a => UU a -> a -> a

diff' :: Num a => UU a -> a -> (a, a)

diffs :: Num a => UU a -> a -> [a]

diffs0 :: Num a => UU a -> a -> [a]

diffsF :: (Functor f, Num a) => UF f a -> a -> f [a]

diffs0F :: (Functor f, Num a) => UF f a -> a -> f [a]

Directional Derivatives

du :: (Functor f, Num a) => FU f a -> f (a, a) -> a

du' :: (Functor f, Num a) => FU f a -> f (a, a) -> (a, a)

dus :: (Functor f, Num a) => FU f a -> f [a] -> [a]

dus0 :: (Functor f, Num a) => FU f a -> f [a] -> [a]

duF :: (Functor f, Functor g, Num a) => FF f g a -> f (a, a) -> g a

duF' :: (Functor f, Functor g, Num a) => FF f g a -> f (a, a) -> g (a, a)

dusF :: (Functor f, Functor g, Num a) => FF f g a -> f [a] -> g [a]

dus0F :: (Functor f, Functor g, Num a) => FF f g a -> f [a] -> g [a]

Monadic Combinators

diffsM :: (Monad m, Num a) => UF m a -> a -> m [a]

diffs0M :: (Monad m, Num a) => UF m a -> a -> m [a]

Exposed Types

type UU a = forall s. Mode s => AD s a -> AD s a

type UF f a = forall s. Mode s => AD s a -> f (AD s a)

type FU f a = forall s. Mode s => f (AD s a) -> AD s a

type FF f g a = forall s. Mode s => f (AD s a) -> g (AD s a)

class Lifted t => Mode t where

Methods

lift :: Num a => a -> t a

Embed a constant

(<+>) :: Num a => t a -> t a -> t a

Vector sum

(*^) :: Num a => a -> t a -> t a

Scalar-vector multiplication

(^*) :: Num a => t a -> a -> t a

Vector-scalar multiplication

(^/) :: Fractional a => t a -> a -> t a

Scalar division

zero :: Num a => t a

 'zero' = 'lift' 0

Instances

Mode Id

Lifted Forward => Mode Forward

Lifted Reverse => Mode Reverse

Lifted Tower => Mode Tower

Mode f => Mode (AD f)

Mode f => Mode (:> f)

(Mode f, Mode g) => Mode (ComposeMode f g)

newtype AD f a

AD serves as a common wrapper for different Mode instances, exposing a traditional numerical tower. Universal quantification is used to limit the actions in user code to machinery that will return the same answers under all AD modes, allowing us to use modes interchangeably as both the type level "brand" and dictionary, providing a common API.

Constructors

AD

Fields

runAD :: f a

Instances

Primal f => Primal (AD f)

Mode f => Mode (AD f)

Lifted f => Lifted (AD f)

Var (AD Reverse)

Iso (f a) (AD f a)

(Num a, Lifted f, Bounded a) => Bounded (AD f a)

(Num a, Lifted f, Enum a) => Enum (AD f a)

(Num a, Lifted f, Eq a) => Eq (AD f a)

(Lifted f, Floating a) => Floating (AD f a)

(Lifted f, Fractional a) => Fractional (AD f a)

(Lifted f, Num a) => Num (AD f a)

(Num a, Lifted f, Ord a) => Ord (AD f a)

(Lifted f, Real a) => Real (AD f a)

(Lifted f, RealFloat a) => RealFloat (AD f a)

(Lifted f, RealFrac a) => RealFrac (AD f a)

(Lifted f, Show a) => Show (AD f a)