Portability | GHC only |
---|---|

Stability | experimental |

Maintainer | ekmett@gmail.com |

Mixed-Mode Automatic Differentiation.

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

- grad :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> f a
- grad' :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f a)
- gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> f b
- gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> (a, f b)
- jacobian :: (Traversable f, Traversable g, Num a) => (forall s. Mode s => f (AD s a) -> g (AD s a)) -> f a -> g (f a)
- 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)
- 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' :: (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)
- diff :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> a
- diffF :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f a
- diff' :: Num a => (forall s. Mode s => AD s a -> AD s a) -> a -> (a, a)
- diffF' :: (Functor f, Num a) => (forall s. Mode s => AD s a -> f (AD s a)) -> a -> f (a, a)
- 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]
- 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 :: 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]
- diffM :: (Monad m, Num a) => (forall s. Mode s => AD s a -> m (AD s a)) -> a -> m a
- diffM' :: (Monad m, Num a) => (forall s. Mode s => AD s a -> m (AD s a)) -> a -> m (a, a)
- 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)
- newtype AD f a = AD {
- runAD :: f a

- class Lifted t => Mode t where

# Gradients

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

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

function calculates the gradient of a non-scalar-to-scalar function `grad`

g f`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)Source

# 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)Source

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)Source

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)Source

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.
`jacobianWith`

g f

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)Source

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.
`jacobianWith'`

g f

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, a)Source

The `d'UU`

function calculates the result and first derivative of scalar-to-scalar function by F`orward`

`AD`

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

# Derivatives (Tower)

# Directional Derivatives (Forward)

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

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)Source

# Taylor Series (Tower)

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

# Monadic Combinators (Forward)

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

The `dUM`

function calculates the first derivative of scalar-to-scalar monadic function by F`orward`

`AD`

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

The `d'UM`

function calculates the result and first derivative of a scalar-to-scalar monadic function by F`orward`

`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)Source

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)Source

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)Source

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)Source

# Exposed Types

`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.

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 whereSource

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' = 'lift' 0