Portability GHC only experimental ekmett@gmail.com

Description

Synopsis

Newton's Method (Forward AD)

findZero :: Fractional a => UU a -> a -> [a]Source

The findZero function finds a zero of a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.)

Examples:

take 10 \$ findZero (\\x->x^2-4) 1  -- converge to 2.0
module Data.Complex
take 10 \$ findZero ((+1).(^2)) (1 :+ 1)  -- converge to (0 :+ 1)@

findZeroM :: (Monad m, Fractional a) => UF m a -> a -> MList m aSource

inverse :: Fractional a => UU a -> a -> a -> [a]Source

The inverseNewton function inverts a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.)

Example:

take 10 \$ inverseNewton sqrt 1 (sqrt 10)  -- converges to 10

inverseM :: (Monad m, Fractional a) => UF m a -> a -> a -> MList m aSource

fixedPoint :: Fractional a => UU a -> a -> [a]Source

The fixedPoint function find a fixedpoint of a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.)

take 10 \$ fixedPoint cos 1 -- converges to 0.7390851332151607

fixedPointM :: (Monad m, Fractional a) => UF m a -> a -> MList m aSource

extremum :: Fractional a => UU a -> a -> [a]Source

The extremum function finds an extremum of a scalar function using Newton's method; produces a stream of increasingly accurate results. (Modulo the usual caveats.)

take 10 \$ extremum cos 1 -- convert to 0

extremumM :: (Monad m, Fractional a) => UF m a -> a -> MList m aSource

gradientDescent :: (Traversable f, Fractional a, Ord a) => FU f a -> f a -> [f a]Source

The gradientDescent function performs a multivariate optimization, based on the naive-gradient-descent in the file stalingrad/examples/flow-tests/pre-saddle-1a.vlad from the VLAD compiler Stalingrad sources. Its output is a stream of increasingly accurate results. (Modulo the usual caveats.)

It uses reverse mode automatic differentiation to compute the gradient.

gradientDescentM :: (Traversable f, Monad m, Fractional a, Ord a) => FF f m a -> f a -> MList m (f a)Source

gradientAscent :: (Traversable f, Fractional a, Ord a) => FU f a -> f a -> [f a]Source

gradientAscentM :: (Traversable f, Monad m, Fractional a, Ord a) => FF f m a -> f a -> MList m (f a)Source

Exposed Types

type UU a = forall s. Mode s => AD s a -> AD s aSource

A scalar-to-scalar automatically-differentiable function.

type UF f a = forall s. Mode s => AD s a -> f (AD s a)Source

A scalar-to-non-scalar automatically-differentiable function.

type FU f a = forall s. Mode s => f (AD s a) -> AD s aSource

A non-scalar-to-scalar automatically-differentiable function.

type FF f g a = forall s. Mode s => f (AD s a) -> g (AD s a)Source

A non-scalar-to-non-scalar automatically-differentiable function.

newtype AD f a Source

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.

Constructors

Instances

class Lifted t => Mode t whereSource

Methods

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 :: Num a => t aSource

'zero' = 'lift' 0

Instances

 Mode Id Lifted Forward => Mode Forward Lifted Reverse => Mode Reverse Lifted Tower => Mode Tower Mode f => Mode (AD f) Mode f => Mode (Iterated f) (Mode f, Mode g) => Mode (ComposeMode f g)