Copyright (c) Edward Kmett 2010-2015 BSD3 ekmett@gmail.com experimental GHC only None Haskell2010

Description

Forward mode automatic differentiation

Synopsis

# Documentation

data Forward a Source

`Forward` mode AD

Instances

 (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) Source 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) Source Erf a => Erf (Forward a) InvErf a => InvErf (Forward a) Num a => Mode (Forward a) Source Num a => Jacobian (Forward a) Source type Scalar (Forward a) = a Source type D (Forward a) = Id a Source

auto :: Mode t => Scalar t -> t Source

Embed a constant

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

Note, this performs O(n) worse than `grad` for `n` inputs, in exchange for better space utilization.

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

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) -> (f (Forward a) -> 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) -> (f (Forward a) -> 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.

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

# Jacobian

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

Compute the Jacobian using `Forward` mode `AD`. This must transpose the result, so `jacobianT` is faster and allows more result types.

````>>> ````jacobian (\[x,y] -> [y,x,x+y,x*y,exp x * sin y]) [pi,1]
```[[0.0,1.0],[1.0,0.0],[1.0,1.0],[1.0,3.141592653589793],[19.472221418841606,12.502969588876512]]
```

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

Compute the Jacobian using `Forward` mode `AD` along with the actual answer.

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

Compute the Jacobian using `Forward` mode `AD` combined with the input using a user specified function, along with the actual answer.

# Transposed Jacobian

jacobianT :: (Traversable f, Functor g, Num a) => (f (Forward a) -> g (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) -> (f (Forward a) -> g (Forward a)) -> f a -> f (g b) Source

A fast, simple, transposed Jacobian computed with `Forward` mode `AD` that combines the output with the input.

# Hessian Product

hessianProduct :: (Traversable f, Num a) => (f (On (Forward (Forward a))) -> 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) => (f (On (Forward (Forward a))) -> On (Forward (Forward a))) -> f (a, a) -> f (a, a) Source

# Derivatives

diff :: Num a => (Forward a -> Forward a) -> a -> a Source

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

````>>> ````diff sin 0
```1.0
```

diff' :: Num a => (Forward a -> Forward a) -> a -> (a, a) Source

The `diff'` function calculates the result and first derivative of scalar-to-scalar function by `Forward` mode `AD`

````diff'` `sin` == `sin` `&&&` `cos`
`diff'` f = f `&&&` d f
```
````>>> ````diff' sin 0
```(0.0,1.0)
```
````>>> ````diff' exp 0
```(1.0,1.0)
```

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

The `diffF` function calculates the first derivatives of scalar-to-nonscalar function by `Forward` mode `AD`

````>>> ````diffF (\a -> [sin a, cos a]) 0
```[1.0,-0.0]
```

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

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

````>>> ````diffF' (\a -> [sin a, cos a]) 0
```[(0.0,1.0),(1.0,-0.0)]
```

# Directional Derivatives

du :: (Functor f, Num a) => (f (Forward a) -> 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) => (f (Forward a) -> 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

duF :: (Functor f, Functor g, Num a) => (f (Forward a) -> g (Forward a)) -> f (a, a) -> g a Source

Compute a vector of directional derivatives for a function given a zipped up `Functor` of the input values and their derivatives.

duF' :: (Functor f, Functor g, Num a) => (f (Forward a) -> g (Forward a)) -> f (a, a) -> g (a, a) Source

Compute a vector of answers and directional derivatives for a function given a zipped up `Functor` of the input values and their derivatives.