Portability GHC only experimental ekmett@gmail.com

Description

Higher order derivatives via a "dual number tower".

Synopsis

# Taylor Series

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

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

# Maclaurin Series

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

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

# Derivatives

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

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

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

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

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

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

# Directional Derivatives

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

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

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

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

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

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

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

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

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

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

# Exposed Types

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

A scalar-to-scalar automatically-differentiable function.

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

A scalar-to-non-scalar automatically-differentiable function.

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

A non-scalar-to-scalar automatically-differentiable function.

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

A non-scalar-to-non-scalar automatically-differentiable function.

class Lifted t => Mode t whereSource

Methods

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

Embed a constant

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

Vector sum

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

Scalar-vector multiplication

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

Vector-scalar multiplication

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

Scalar division

zero :: Num a => t aSource

``` 'zero' = 'lift' 0
```

Instances

 Mode Id Lifted Forward => Mode Forward Lifted Tower => Mode Tower Lifted Reverse => Mode Reverse Lifted Sparse => Mode Sparse Mode f => Mode (AD f) (Traversable f, Lifted (Dense f)) => Mode (Dense f) (Mode f, Mode g) => Mode (ComposeMode f g)

`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.