Copyright | (c) Edward Kmett 2010-2014 |
---|---|

License | BSD3 |

Maintainer | ekmett@gmail.com |

Stability | experimental |

Portability | GHC only |

Safe Haskell | None |

Language | Haskell2010 |

Forward mode automatic differentiation

- data AD s a
- data Forward a
- auto :: Mode t => Scalar t -> t
- grad :: (Traversable f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f a -> f a
- grad' :: (Traversable f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f a -> (a, f a)
- gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f a -> f b
- gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f a -> (a, f b)
- jacobian :: (Traversable f, Traversable g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> g (f a)
- jacobian' :: (Traversable f, Traversable g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> g (a, f a)
- jacobianWith :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> g (f b)
- jacobianWith' :: (Traversable f, Traversable g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> g (a, f b)
- jacobianT :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> f (g a)
- jacobianWithT :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> f (g b)
- hessianProduct :: (Traversable f, Num a) => (forall s. f (AD s (On (Forward (Forward a)))) -> AD s (On (Forward (Forward a)))) -> f (a, a) -> f a
- hessianProduct' :: (Traversable f, Num a) => (forall s. f (AD s (On (Forward (Forward a)))) -> AD s (On (Forward (Forward a)))) -> f (a, a) -> f (a, a)
- diff :: Num a => (forall s. AD s (Forward a) -> AD s (Forward a)) -> a -> a
- diff' :: Num a => (forall s. AD s (Forward a) -> AD s (Forward a)) -> a -> (a, a)
- diffF :: (Functor f, Num a) => (forall s. AD s (Forward a) -> f (AD s (Forward a))) -> a -> f a
- diffF' :: (Functor f, Num a) => (forall s. AD s (Forward a) -> f (AD s (Forward a))) -> a -> f (a, a)
- du :: (Functor f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f (a, a) -> a
- du' :: (Functor f, Num a) => (forall s. f (AD s (Forward a)) -> AD s (Forward a)) -> f (a, a) -> (a, a)
- duF :: (Functor f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f (a, a) -> g a
- duF' :: (Functor f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f (a, a) -> g (a, a)

# Documentation

Bounded a => Bounded (AD s a) | |

Enum a => Enum (AD s a) | |

Eq a => Eq (AD s a) | |

Floating a => Floating (AD s a) | |

Fractional a => Fractional (AD s a) | |

Num a => Num (AD s a) | |

Ord a => Ord (AD s a) | |

Read a => Read (AD s a) | |

Real a => Real (AD s a) | |

RealFloat a => RealFloat (AD s a) | |

RealFrac a => RealFrac (AD s a) | |

Show a => Show (AD s a) | |

Erf a => Erf (AD s a) | |

InvErf a => InvErf (AD s a) | |

Mode a => Mode (AD s a) | |

Typeable (* -> * -> *) AD | |

type Scalar (AD s a) = Scalar a |

`Forward`

mode AD

(Num a, Bounded a) => Bounded (Forward a) | |

(Num a, Enum a) => Enum (Forward a) | |

(Num a, Eq a) => Eq (Forward a) | |

Floating a => Floating (Forward a) | |

Fractional a => Fractional (Forward a) | |

Data a => Data (Forward a) | |

Num a => Num (Forward a) | |

(Num a, Ord a) => Ord (Forward a) | |

Real a => Real (Forward a) | |

RealFloat a => RealFloat (Forward a) | |

RealFrac a => RealFrac (Forward a) | |

Show a => Show (Forward a) | |

Erf a => Erf (Forward a) | |

InvErf a => InvErf (Forward a) | |

Num a => Mode (Forward a) | |

Num a => Jacobian (Forward a) | |

Typeable (* -> *) Forward | |

type Scalar (Forward a) = a | |

type D (Forward a) = Id a |

# Gradient

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

Compute the gradient of a function using forward mode AD.

Note, this performs *O(n)* worse than `grad`

for `n`

inputs, in exchange for better space utilization.

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

Compute the gradient and answer to a function using forward mode AD.

Note, this performs *O(n)* worse than `grad'`

for `n`

inputs, in exchange for better space utilization.

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

Compute the gradient of a function using forward mode AD and combine the result with the input using a user-specified function.

Note, this performs *O(n)* worse than `gradWith`

for `n`

inputs, in exchange for better space utilization.

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

Compute the gradient of a function using forward mode AD and the answer, and combine the result with the input using a user-specified function.

Note, this performs *O(n)* worse than `gradWith'`

for `n`

inputs, in exchange for better space utilization.

`>>>`

(10,[(0,1),(1,1),(2,1),(3,1),(4,1)])`gradWith' (,) sum [0..4]`

# Jacobian

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

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

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

Compute the Jacobian using `Forward`

mode `AD`

and combine the output with the input. This must transpose the result, so `jacobianWithT`

is faster, and allows more result types.

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

# Transposed Jacobian

jacobianT :: (Traversable f, Functor g, Num a) => (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> f (g a) Source

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

jacobianWithT :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (forall s. f (AD s (Forward a)) -> g (AD s (Forward a))) -> f a -> f (g b) Source

# Hessian Product

hessianProduct :: (Traversable f, Num a) => (forall s. f (AD s (On (Forward (Forward a)))) -> AD s (On (Forward (Forward a)))) -> f (a, a) -> f a Source

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

hessianProduct' :: (Traversable f, Num a) => (forall s. f (AD s (On (Forward (Forward a)))) -> AD s (On (Forward (Forward a)))) -> f (a, a) -> f (a, a) Source

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

# Derivatives

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

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

# Directional Derivatives

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

Compute the directional derivative of a function given a zipped up `Functor`

of the input values and their derivatives

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

Compute the answer and directional derivative of a function given a zipped up `Functor`

of the input values and their derivatives