Portability | GHC only |
---|---|
Stability | experimental |
Maintainer | ekmett@gmail.com |
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 => 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]
- 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 => 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 => 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 => 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 => 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
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