Numeric.AD.Newton.Double

Newton's Method (Forward AD)

findZero :: (forall s. AD s ForwardDouble -> AD s ForwardDouble) -> Double -> [Double]

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]

findZeroNoEq :: (forall s. AD s ForwardDouble -> AD s ForwardDouble) -> Double -> [Double]

The findZeroNoEq function behaves the same as findZero except that it doesn't truncate the list once the results become constant.

inverse :: (forall s. AD s ForwardDouble -> AD s ForwardDouble) -> Double -> Double -> [Double]

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 :: (forall s. AD s ForwardDouble -> AD s ForwardDouble) -> Double -> Double -> [Double]

The inverseNoEq function behaves the same as inverse except that it doesn't truncate the list once the results become constant.

fixedPoint :: (forall s. AD s ForwardDouble -> AD s ForwardDouble) -> Double -> [Double]

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 :: (forall s. AD s ForwardDouble -> AD s ForwardDouble) -> Double -> [Double]

The fixedPointNoEq function behaves the same as fixedPoint except that doesn't truncate the list once the results become constant.

extremum :: (forall s. AD s (On (Forward ForwardDouble)) -> AD s (On (Forward ForwardDouble))) -> Double -> [Double]

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 :: (forall s. AD s (On (Forward ForwardDouble)) -> AD s (On (Forward ForwardDouble))) -> Double -> [Double]

The extremumNoEq function behaves the same as extremum except that it doesn't truncate the list once the results become constant.

Gradient Ascent/Descent (Reverse AD)

conjugateGradientDescent :: Traversable f => (forall s. Chosen s => f (Or s (On (Forward ForwardDouble)) (Kahn Double)) -> Or s (On (Forward ForwardDouble)) (Kahn Double)) -> f Double -> [f Double]

Perform a conjugate gradient descent using reverse mode automatic differentiation to compute the gradient, and using forward-on-forward mode for computing extrema.

>>> let sq x = x * x
>>> let rosenbrock [x,y] = sq (1 - x) + 100 * sq (y - sq x)
>>> rosenbrock [0,0]
1
>>> rosenbrock (conjugateGradientDescent rosenbrock [0, 0] !! 5) < 0.1
True

conjugateGradientAscent :: Traversable f => (forall s. Chosen s => f (Or s (On (Forward ForwardDouble)) (Kahn Double)) -> Or s (On (Forward ForwardDouble)) (Kahn Double)) -> f Double -> [f Double]

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