ad-0.45.0: Automatic Differentiation

PortabilityGHC only
Stabilityexperimental
Maintainerekmett@gmail.com

Numeric.AD.Newton

Contents

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)@

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

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

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 

Gradient Ascent/Descent (Reverse AD)

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.

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

AD 

Fields

runAD :: f a
 

Instances

Typeable1 f => Typeable1 (AD f) 
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) 
(Typeable1 f, Typeable a, Data (f a), Data a) => Data (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) 
Num a => Grad (AD Reverse a) [a] (a, [a]) a 
Num a => Grad (AD Sparse a) [a] (a, [a]) a 
Grads i o a => Grads (AD Sparse a -> i) (a -> o) a 
Num a => Grads (AD Sparse a) (Stream [] a) a 
Grad i o o' a => Grad (AD Reverse a -> i) (a -> o) (a -> o') a 
Grad i o o' a => Grad (AD Sparse a -> i) (a -> o) (a -> o') a 

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