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

Stability | experimental |

Maintainer | ekmett@gmail.com |

Safe Haskell | Safe-Infered |

Root finding using Halley's rational method (the second in the class of Householder methods). Assumes the function is three times continuously differentiable and converges cubically when progress can be made.

- findZero :: (Fractional a, Eq a) => UU a -> a -> [a]
- inverse :: (Fractional a, Eq a) => UU a -> a -> a -> [a]
- fixedPoint :: (Fractional a, Eq a) => UU a -> a -> [a]
- extremum :: (Fractional a, Eq a) => UU a -> a -> [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

# Halley's Method (Tower AD)

findZero :: (Fractional a, Eq a) => UU a -> a -> [a]Source

The `findZero`

function finds a zero of a scalar function using
Halley'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, Eq a) => UU a -> a -> a -> [a]Source

The `inverse`

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

Note: the `take 10 $ inverse sqrt 1 (sqrt 10)`

example that works for Newton's method
fails with Halley's method because the preconditions do not hold.

fixedPoint :: (Fractional a, Eq a) => UU a -> a -> [a]Source

The `fixedPoint`

function find a fixedpoint of a scalar
function using Halley'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, Eq a) => UU a -> a -> [a]Source

The `extremum`

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

take 10 $ extremum cos 1 -- convert to 0

# 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

isKnownConstant :: t a -> BoolSource

allowed to return False for items with a zero derivative, but we'll give more NaNs than strictly necessary

isKnownZero :: Num a => t a -> BoolSource

allowed to return False for zero, but we give more NaN's than strictly necessary then

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

(<**>) :: Floating a => t a -> t a -> t aSource

Exponentiation, this should be overloaded if you can figure out anything about what is constant!

'zero' = 'lift' 0