Portability  GHC only 

Stability  experimental 
Maintainer  ekmett@gmail.com 
Safe Haskell  SafeInfered 
 Gradients (Reverse Mode)
 Higher Order Gradients (SparseonReverse)
 Jacobians (Sparse or Reverse)
 Higher Order Jacobian (SparseonReverse)
 Transposed Jacobians (Forward Mode)
 Hessian (SparseOnReverse)
 Hessian Tensors (Sparse or SparseOnReverse)
 Hessian Tensors (Sparse)
 Hessian Vector Products (ForwardOnReverse)
 Derivatives (Forward Mode)
 Derivatives (Tower)
 Directional Derivatives (Forward Mode)
 Directional Derivatives (Tower)
 Taylor Series (Tower)
 Maclaurin Series (Tower)
MixedMode Automatic Differentiation.
Each combinator exported from this module chooses an appropriate AD mode. The following basic operations are supported, modified as appropriate by the suffixes below:

grad
computes the gradient (partial derivatives) of a function at a point 
jacobian
computes the Jacobian matrix of a function at a point 
diff
computes the derivative of a function at a point 
du
computes a directional derivative of a function at a point 
hessian
compute the Hessian matrix (matrix of second partial derivatives) of a function at a point
The suffixes have the following meanings:

'
 also return the answer 
With
lets the user supply a function to blend the input with the output 
F
is a version of the base function lifted to return aTraversable
(orFunctor
) result 
s
means the function returns all higher derivatives in a list or fbranchingStream

T
means the result is transposed with respect to the traditional formulation. 
0
means that the resulting derivative list is padded with 0s at the end.
 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)
 grads :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) > AD s a) > f a > Cofree f a
 jacobian :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > g (f a)
 jacobian' :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > g (a, f a)
 jacobianWith :: (Traversable f, Functor 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, Functor g, Num a) => (a > a > b) > (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > g (a, f b)
 jacobians :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > g (Cofree f a)
 jacobianT :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > f (g a)
 jacobianWithT :: (Traversable f, Functor g, Num a) => (a > a > b) > (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > f (g b)
 hessian :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) > AD s a) > f a > f (f a)
 hessian' :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) > AD s a) > f a > (a, f (a, f a))
 hessianF :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > g (f (f a))
 hessianF' :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > g (a, f (a, f a))
 hessianProduct :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) > AD s a) > f (a, a) > f a
 hessianProduct' :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) > AD s a) > f (a, a) > f (a, a)
 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)
 dus :: (Functor f, Num a) => (forall s. Mode s => f (AD s a) > AD s a) > f [a] > [a]
 dus0 :: (Functor f, Num a) => (forall s. Mode s => f (AD s a) > AD s a) > f [a] > [a]
 dusF :: (Functor f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f [a] > g [a]
 dus0F :: (Functor f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f [a] > g [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]
Gradients (Reverse Mode)
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 nonscalartoscalar function grad
g ff
with reversemode AD in a single pass.
The gradient is combined elementwise 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
Higher Order Gradients (SparseonReverse)
grads :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) > AD s a) > f a > Cofree f aSource
Jacobians (Sparse or Reverse)
jacobian :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > g (f a)Source
jacobian' :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > g (a, f a)Source
jacobianWith :: (Traversable f, Functor 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 nonscalartononscalar 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 elementwise with the input using g
.
If you know the relative number of inputs and outputs, consider jacobianWith
or jacobianWith
.
jacobianWith' :: (Traversable f, Functor 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 nonscalartononscalar function, automatically choosing between sparse and reverse mode AD based on the number of inputs and outputs.
jacobianWith'
g f
The resulting Jacobian matrix is then recombined elementwise with the input using g
.
If you know the relative number of inputs and outputs, consider jacobianWith'
or jacobianWith'
.
Higher Order Jacobian (SparseonReverse)
jacobians :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > g (Cofree f a)Source
Transposed Jacobians (Forward Mode)
jacobianT :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > f (g a)Source
A fast, simple transposed Jacobian computed with forwardmode AD.
jacobianWithT :: (Traversable f, Functor g, Num a) => (a > a > b) > (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > f (g b)Source
A fast, simple transposed Jacobian computed with forwardmode AD.
Hessian (SparseOnReverse)
hessian :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) > AD s a) > f a > f (f a)Source
Compute the Hessian via the Jacobian of the gradient. gradient is computed in reverse mode and then the Jacobian is computed in sparse (forward) mode.
hessian' :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) > AD s a) > f a > (a, f (a, f a))Source
Hessian Tensors (Sparse or SparseOnReverse)
hessianF :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > g (f (f a))Source
Compute the order 3 Hessian tensor on a nonscalartononscalar function using Sparse or SparseonReverse
Hessian Tensors (Sparse)
hessianF' :: (Traversable f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f a > g (a, f (a, f a))Source
Hessian Vector Products (ForwardOnReverse)
hessianProduct :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) > AD s a) > f (a, a) > f aSource
computes the product of the hessian hessianProduct
f wvH
of a nonscalartoscalar function f
at w =
with a vector fst
$ wvv = snd $ wv
using "Pearlmutter's method" from http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.6143, which states:
H v = (d/dr) grad_w (w + r v)  r = 0
Or in other words, we take the directional derivative of the gradient. The gradient is calculated in reverse mode, then the directional derivative is calculated in forward mode.
hessianProduct' :: (Traversable f, Num a) => (forall s. Mode s => f (AD s a) > AD s a) > f (a, a) > f (a, a)Source
computes both the gradient of a nonscalartoscalar hessianProduct'
f wvf
at w =
and the product of the hessian fst
$ wvH
at w
with a vector v = snd $ wv
using "Pearlmutter's method". The outputs are returned wrapped in the same functor.
H v = (d/dr) grad_w (w + r v)  r = 0
Or in other words, we return the gradient and the directional derivative of the gradient. The gradient is calculated in reverse mode, then the directional derivative is calculated in forward mode.
Derivatives (Forward Mode)
diff' :: Num a => (forall s. Mode s => AD s a > AD s a) > a > (a, a)Source
The d'
function calculates the result and first derivative of scalartoscalar function by Forward
AD
d' sin == sin &&& cos d' f = f &&& d f
Derivatives (Tower)
Directional Derivatives (Forward Mode)
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
Directional Derivatives (Tower)
dusF :: (Functor f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f [a] > g [a]Source
dus0F :: (Functor f, Functor g, Num a) => (forall s. Mode s => f (AD s a) > g (AD s a)) > f [a] > g [a]Source
Taylor Series (Tower)
Maclaurin Series (Tower)
maclaurin0 :: Fractional a => (forall s. Mode s => AD s a > AD s a) > a > [a]Source