Numeric.AD

Mixed-Mode Automatic Differentiation.

Each combinator exported from this module chooses an appropriate AD mode.

Gradients

grad :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> f a

The grad function calculates the gradient of a non-scalar-to-scalar function with Reverse AD in a single pass.

grad' :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f a)

The grad' function calculates the result and gradient of a non-scalar-to-scalar function with Reverse AD in a single pass.

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> f b

grad g f function calculates the gradient of a non-scalar-to-scalar function f with reverse-mode AD in a single pass. The gradient is combined element-wise with the argument using the function g.

 grad == gradWith (\_ dx -> dx)
 id == gradWith const

gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f b)

grad' g f calculates the result and gradient of a non-scalar-to-scalar function f with Reverse AD in a single pass the gradient is combined element-wise with the argument using the function g.

 grad' == gradWith' (\_ dx -> dx)

Jacobians

jacobian :: (Traversable f, Traversable g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f a)

Calculate the Jacobian of a non-scalar-to-non-scalar function, automatically choosing between forward and reverse mode AD based on the number of inputs and outputs.

If you need to support functions where the output is only a Functor or Monad, consider Numeric.AD.Reverse.jacobian or gradM from Numeric.AD.Reverse.

jacobian' :: (Traversable f, Traversable g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (a, f a)

Calculate both the answer and Jacobian of a non-scalar-to-non-scalar function, automatically choosing between forward- and reverse- mode AD based on the relative, number of inputs and outputs.

If you need to support functions where the output is only a Functor or Monad, consider Numeric.AD.Reverse.jacobian' or gradM' from Numeric.AD.Reverse.

jacobianWith :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f b)

jacobianWith g f calculates the Jacobian of a non-scalar-to-non-scalar function, automatically choosing between forward and reverse mode AD based on the number of inputs and outputs.

The resulting Jacobian matrix is then recombined element-wise with the input using g.

If you need to support functions where the output is only a Functor or Monad, consider Numeric.AD.Reverse.jacobianWith or gradWithM from Numeric.AD.Reverse.

jacobianWith' :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (a, f b)

jacobianWith' g f calculates the answer and Jacobian of a non-scalar-to-non-scalar function, automatically choosing between forward and reverse mode AD based on the number of inputs and outputs.

The resulting Jacobian matrix is then recombined element-wise with the input using g.

If you need to support functions where the output is only a Functor or Monad, consider Numeric.AD.Reverse.jacobianWith' or gradWithM' from Numeric.AD.Reverse.

Derivatives (Forward)

diff :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> a

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

 diff sin == cos

diffF :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f a

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

diff' :: Num a => (forall s. Mode s => AD s a -> AD s 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) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f (a, a)

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

Derivatives (Tower)

diffs :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]

diffsF :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f [a]

diffs0 :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]

diffs0F :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f [a]

Directional Derivatives (Forward)

du :: (Functor f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f (a, a) -> a

du' :: (Functor f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f (a, a) -> (a, a)

duF :: (Functor f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f (a, a) -> g a

duF' :: (Functor f, Functor g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f (a, a) -> g (a, a)

Taylor Series (Tower)

taylor :: Fractional a => (forall s. Mode s => AD s a -> AD s a) -> a -> a -> [a]

taylor0 :: Fractional a => (forall s. Mode s => AD s a -> AD s a) -> a -> a -> [a]

maclaurin :: Fractional a => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]

maclaurin0 :: Fractional a => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]

Monadic Combinators (Forward)

diffM :: (Monad m, Num a) => (forall s. Mode s => AD s a -> m (AD s a)) -> a -> m a

The dUM function calculates the first derivative of scalar-to-scalar monadic function by Forward AD

diffM' :: (Monad m, Num a) => (forall s. Mode s => AD s a -> m (AD s a)) -> a -> m (a, a)

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

Monadic Combinators (Reverse)

gradM :: (Traversable f, Monad m, Num a) => (forall s. Mode s => f (AD s a) -> m (AD s a)) -> f a -> m (f a)

gradM' :: (Traversable f, Monad m, Num a) => (forall s. Mode s => f (AD s a) -> m (AD s a)) -> f a -> m (a, f a)

gradWithM :: (Traversable f, Monad m, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> m (AD s a)) -> f a -> m (f b)

gradWithM' :: (Traversable f, Monad m, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> m (AD s a)) -> f a -> m (a, f b)

Exposed Types

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)

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)