Numeric.AD.Mode.Forward

Forward mode automatic differentiation

Gradient

grad :: (Traversable f, Num a) => FU f a -> f a -> f a

grad' :: (Traversable f, Num a) => FU f a -> f a -> (a, f a)

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> FU f a -> f a -> f b

gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> FU f a -> f a -> (a, f b)

Jacobian

jacobian :: (Traversable f, Traversable g, Num a) => FF f g a -> f a -> g (f a)

jacobian' :: (Traversable f, Traversable g, Num a) => FF f g a -> f a -> g (a, f a)

jacobianWith :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (f b)

jacobianWith' :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> FF f g a -> f a -> g (a, f b)

Transposed Jacobian

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

A fast, simple transposed Jacobian computed with forward-mode AD.

jacobianWithT :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> FF f g a -> f a -> f (g b)

A fast, simple transposed Jacobian computed with forward-mode AD.

Hessian Product

hessianProduct :: (Traversable f, Num a) => FU f a -> f (a, a) -> f a

Compute the product of a vector with the Hessian using forward-on-forward-mode AD.

hessianProduct' :: (Traversable f, Num a) => FU f a -> f (a, a) -> f (a, a)

Compute the gradient and hessian product using forward-on-forward-mode AD.

Derivatives

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

The diff function calculates the first derivative of a scalar-to-scalar function by forward-mode AD

 diff sin == cos

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

The d'UU function calculates the result and first derivative of scalar-to-scalar function by Forward AD

 d' sin == sin &&& cos
 d' f = f &&& d f

diffF :: (Functor f, Num a) => UF f a -> a -> f a

The diffF function calculates the first derivative of scalar-to-nonscalar function by Forward AD

diffF' :: (Functor f, Num a) => UF f a -> a -> f (a, a)

The diffF' function calculates the result and first derivative of a scalar-to-non-scalar function by Forward AD

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)

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)

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.

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

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 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)