Numeric.AD.Newton

Newton's Method (Forward AD)

findZero :: Fractional a => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]

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)@

inverse :: Fractional a => (forall s. Mode s => AD s a -> AD s a) -> a -> a -> [a]

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)  -- converge to 10

fixedPoint :: Fractional a => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]

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.)

extremum :: Fractional a => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]

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.)

Gradient Descent (Reverse AD)

gradientDescent :: (Traversable f, Fractional a, Ord a) => (forall s. Mode s => f (AD s a) -> AD s a) -> f a -> [f a]

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.

Exposed Types

newtype AD f a

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

AD

Fields

runAD :: f a

Instances

Primal f => Primal (AD f)

Mode f => Mode (AD f)

Lifted f => Lifted (AD f)

Var (AD Reverse)

Iso (f a) (AD f a)

(Num a, Lifted f, Bounded a) => Bounded (AD f a)

(Num a, Lifted f, Enum a) => Enum (AD f a)

(Num a, Lifted f, Eq a) => Eq (AD f a)

(Lifted f, Floating a) => Floating (AD f a)

(Lifted f, Fractional a) => Fractional (AD f a)

(Lifted f, Num a) => Num (AD f a)

(Num a, Lifted f, Ord a) => Ord (AD f a)

(Lifted f, Real a) => Real (AD f a)

(Lifted f, RealFloat a) => RealFloat (AD f a)

(Lifted f, RealFrac a) => RealFrac (AD f a)

(Lifted f, Show a) => Show (AD f a)

class Lifted t => Mode t where

Methods

lift :: Num a => a -> t a

Embed a constant

(<+>) :: Num a => t a -> t a -> t a

Vector sum

(*^) :: Num a => a -> t a -> t a

Scalar-vector multiplication

(^*) :: Num a => t a -> a -> t a

Vector-scalar multiplication

(^/) :: Fractional a => t a -> a -> t a

Scalar division

zero :: Num a => t a

 '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 g) => Mode (:. f g)