Numeric.AD.Rank1.Newton

Newton's Method (Forward)

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

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.) If the stream becomes constant ("it converges"), no further elements are returned.

Examples:

>>> take 10 $ findZero (\x->x^2-4) 1
[1.0,2.5,2.05,2.000609756097561,2.0000000929222947,2.000000000000002,2.0]

>>> last $ take 10 $ findZero ((+1).(^2)) (1 :+ 1)
0.0 :+ 1.0

findZeroNoEq :: Fractional a => (Forward a -> Forward a) -> a -> [a]

The findZeroNoEq function behaves the same as findZero except that it doesn't truncate the list once the results become constant. This means it can be used with types without an Eq instance.

inverse :: (Fractional a, Eq a) => (Forward a -> Forward a) -> a -> a -> [a]

The inverse function inverts a scalar function using Newton's method; its output is a stream of increasingly accurate results. (Modulo the usual caveats.) If the stream becomes constant ("it converges"), no further elements are returned.

Example:

>>> last $ take 10 $ inverse sqrt 1 (sqrt 10)
10.0

inverseNoEq :: Fractional a => (Forward a -> Forward a) -> a -> a -> [a]

The inverseNoEq function behaves the same as inverse except that it doesn't truncate the list once the results become constant. This means it can be used with types without an Eq instance.

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

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.)

If the stream becomes constant ("it converges"), no further elements are returned.

>>> last $ take 10 $ fixedPoint cos 1
0.7390851332151607

fixedPointNoEq :: Fractional a => (Forward a -> Forward a) -> a -> [a]

The fixedPointNoEq function behaves the same as fixedPoint except that it doesn't truncate the list once the results become constant. This means it can be used with types without an Eq instance.

extremum :: (Fractional a, Eq a) => (On (Forward (Forward a)) -> On (Forward (Forward a))) -> a -> [a]

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.) If the stream becomes constant ("it converges"), no further elements are returned.

>>> last $ take 10 $ extremum cos 1
0.0

extremumNoEq :: Fractional a => (On (Forward (Forward a)) -> On (Forward (Forward a))) -> a -> [a]

The extremumNoEq function behaves the same as extremum except that it doesn't truncate the list once the results become constant. This means it can be used with types without an Eq instance.

Gradient Ascent/Descent (Kahn)

gradientDescent :: (Traversable f, Fractional a, Ord a) => (f (Kahn a) -> Kahn a) -> f a -> [f a]

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) => (f (Kahn a) -> Kahn a) -> f a -> [f a]

Perform a gradient descent using reverse mode automatic differentiation to compute the gradient.