Numeric.AD.Mode.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]

Exposed Types

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

A scalar-to-scalar automatically-differentiable function.

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

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

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

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)

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

class Lifted t => Mode t where

Methods

isKnownConstant :: t a -> Bool

allowed to return False for items with a zero derivative, but we'll give more NaNs than strictly necessary

isKnownZero :: Num a => t a -> Bool

allowed to return False for zero, but we give more NaN's than strictly necessary then

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

(<**>) :: Floating a => t a -> t a -> t a

Exponentiation, this should be overloaded if you can figure out anything about what is constant!

zero :: Num a => t a

 '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)

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

Typeable1 f => Typeable1 (AD f)

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)

(Typeable1 f, Typeable a, Data (f a), Data a) => Data (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)

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

Num a => Grad (AD Reverse a) [a] (a, [a]) a

Num a => Grad (AD Sparse a) [a] (a, [a]) a

Grads i o a => Grads (AD Sparse a -> i) (a -> o) a

Num a => Grads (AD Sparse a) (Cofree [] a) a

Grad i o o' a => Grad (AD Reverse a -> i) (a -> o) (a -> o') a

Grad i o o' a => Grad (AD Sparse a -> i) (a -> o) (a -> o') a