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.

grad2 :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f a)

The grad2 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

gradWith2 :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f b)

grad2 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.

 grad2 == gradWith2 (\_ 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

jacobian2 :: (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 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, consider using jacobianT from Numeric.AD.Forward or jacobian2 from Numeric.AD.Reverse directly.

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.

jacobianWith2 :: (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)

jacobianWith2 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.

Synonyms

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

The diff2 function calculates the first derivative of scalar-to-scalar function by Forward AD

diff2 :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> (a, a)

The diff2 function calculates the result and first derivative of scalar-to-scalar function by Forward AD

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

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

Derivatives (Forward)

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

The diffUU function calculates the first derivative of a scalar-to-scalar function by Forward AD

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

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

diff2UU :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> (a, a)

The diff2UU function calculates the result and first derivative of scalar-to-scalar function by Forward AD

diff2UF :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f (a, a)

The diff2UF function calculates the result and first derivative of a scalar-to-non-scalar function by Forward AD

Derivatives (Reverse)

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

diff2FU :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f a)

Derivatives (Tower)

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

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

diffs0UU :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]

diffs0UF :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f [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]

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)