Numeric.AD.Rank1.Sparse

Higher order derivatives via a "dual number tower".

Documentation

data Sparse a

We only store partials in sorted order, so the map contained in a partial will only contain partials with equal or greater keys to that of the map in which it was found. This should be key for efficiently computing sparse hessians. there are only (n + k - 1) choose k distinct nth partial derivatives of a function with k inputs.

Instances

Typeable1 Sparse

(Num a, Bounded a) => Bounded (Sparse a)

(Num a, Enum a) => Enum (Sparse a)

(Num a, Eq a) => Eq (Sparse a)

Floating a => Floating (Sparse a)

Fractional a => Fractional (Sparse a)

Data a => Data (Sparse a)

Num a => Num (Sparse a)

(Num a, Ord a) => Ord (Sparse a)

Real a => Real (Sparse a)

RealFloat a => RealFloat (Sparse a)

RealFrac a => RealFrac (Sparse a)

Show a => Show (Sparse a)

Erf a => Erf (Sparse a)

InvErf a => InvErf (Sparse a)

Num a => Mode (Sparse a)

Num a => Jacobian (Sparse a)

Num a => Grad (Sparse a) [a] (a, [a]) a

Num a => Grads (Sparse a) (Cofree [] a) a

Grads i o a => Grads (Sparse a -> i) (a -> o) a

Grad i o o' a => Grad (Sparse a -> i) (a -> o) (a -> o') a

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

Embed a constant

Sparse Gradients

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

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

gradWith :: (Traversable f, Num a) => (a -> a -> b) -> (f (Sparse a) -> Sparse a) -> f a -> f b

gradWith' :: (Traversable f, Num a) => (a -> a -> b) -> (f (Sparse a) -> Sparse a) -> f a -> (a, f b)

Variadic Gradients

Variadic combinators for variadic mixed-mode automatic differentiation.

Unfortunately, variadicity comes at the expense of being able to use quantification to avoid sensitivity confusion, so be careful when counting the number of auto calls you use when taking the gradient of a function that takes gradients!

class Num a => Grad i o o' a | i -> a o o', o -> a i o', o' -> a i o

Instances

Num a => Grad (Sparse a) [a] (a, [a]) a

Grad i o o' a => Grad (Sparse a -> i) (a -> o) (a -> o') a

vgrad :: Grad i o o' a => i -> o

Higher-Order Gradients

grads :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> Cofree f a

Variadic Higher-Order Gradients

class Num a => Grads i o a | i -> a o, o -> a i

Instances

Num a => Grads (Sparse a) (Cofree [] a) a

Grads i o a => Grads (Sparse a -> i) (a -> o) a

vgrads :: Grads i o a => i -> o

Sparse Jacobians (synonyms)

jacobian :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (f a)

jacobian' :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (a, f a)

jacobianWith :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (f (Sparse a) -> g (Sparse a)) -> f a -> g (f b)

jacobianWith' :: (Traversable f, Functor g, Num a) => (a -> a -> b) -> (f (Sparse a) -> g (Sparse a)) -> f a -> g (a, f b)

jacobians :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (Cofree f a)

Sparse Hessians

hessian :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> f (f a)

hessian' :: (Traversable f, Num a) => (f (Sparse a) -> Sparse a) -> f a -> (a, f (a, f a))

hessianF :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (f (f a))

hessianF' :: (Traversable f, Functor g, Num a) => (f (Sparse a) -> g (Sparse a)) -> f a -> g (a, f (a, f a))