Portability | GHC only |
---|---|

Stability | experimental |

Maintainer | ekmett@gmail.com |

- findZero :: Fractional a => UU a -> a -> [a]
- inverse :: Fractional a => UU a -> a -> a -> [a]
- fixedPoint :: Fractional a => UU a -> a -> [a]
- extremum :: Fractional a => UU a -> a -> [a]
- gradientDescent :: (Traversable f, Fractional a, Ord a) => FU f a -> f a -> [f a]
- gradientAscent :: (Traversable f, Fractional a, Ord a) => FU f a -> f a -> [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

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.

`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) | |

(Num 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) (Cofree [] 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

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