Portability | GHC only |
---|---|
Stability | experimental |
Maintainer | ekmett@gmail.com |
- findZero :: Fractional a => UU a -> a -> [a]
- findZeroM :: (Monad m, Fractional a) => UF m a -> a -> MList m a
- inverse :: Fractional a => UU a -> a -> a -> [a]
- inverseM :: (Monad m, Fractional a) => UF m a -> a -> a -> MList m a
- fixedPoint :: Fractional a => UU a -> a -> [a]
- fixedPointM :: (Monad m, Fractional a) => UF m a -> a -> MList m a
- extremum :: Fractional a => UU a -> a -> [a]
- extremumM :: (Monad m, Fractional a) => UF m a -> a -> MList m a
- gradientDescent :: (Traversable f, Fractional a, Ord a) => FU f a -> f a -> [f a]
- gradientDescentM :: (Traversable f, Monad m, Fractional a, Ord a) => FF f m a -> f a -> MList m (f a)
- gradientAscent :: (Traversable f, Fractional a, Ord a) => FU f a -> f a -> [f a]
- gradientAscentM :: (Traversable f, Monad m, Fractional a, Ord a) => FF f m a -> f a -> MList m (f a)
- type UU a = forall s. Mode s => AD s a -> AD s a
- type UF f a = forall s. Mode s => AD s a -> f (AD s a)
- type FU f a = forall s. Mode s => f (AD s a) -> AD s a
- type FF f g a = forall s. Mode s => f (AD s a) -> g (AD s a)
- newtype AD f a = AD {
- runAD :: f a
- class Lifted t => Mode t where
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
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
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.
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.
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.
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) | |
(Typeable1 f, Typeable a) => Typeable (AD f a) |
class Lifted t => Mode t whereSource
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' = 'lift' 0