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

Stability | experimental |

Maintainer | ekmett@gmail.com |

Safe Haskell | None |

- findZero :: (Fractional a, Eq a) => (forall s. Forward a s -> Forward a s) -> a -> [a]
- inverse :: (Fractional a, Eq a) => (forall s. Forward a s -> Forward a s) -> a -> a -> [a]
- fixedPoint :: (Fractional a, Eq a) => (forall s. Forward a s -> Forward a s) -> a -> [a]
- extremum :: (Fractional a, Eq a) => (forall s s'. On (Forward (Forward a s') s) -> On (Forward (Forward a s') s)) -> a -> [a]
- gradientDescent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => f (Reverse a s) -> Reverse a s) -> f a -> [f a]
- gradientAscent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => f (Reverse a s) -> Reverse a s) -> f a -> [f a]
- conjugateGradientDescent :: (Traversable f, Ord a, Fractional a) => (forall t. (Mode t, a ~ Scalar t, Num t) => f t -> t) -> f a -> [f a]
- conjugateGradientAscent :: (Traversable f, Ord a, Fractional a) => (forall t. (Mode t, a ~ Scalar t, Num t) => f t -> t) -> f a -> [f a]

# Newton's Method (Forward AD)

findZero :: (Fractional a, Eq a) => (forall s. Forward a s -> Forward a s) -> 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.) If the stream becomes constant
(it converges), no further elements are returned.

Examples:

`>>>`

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

`>>>`

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

inverse :: (Fractional a, Eq a) => (forall s. Forward a s -> Forward a s) -> a -> a -> [a]Source

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:

`>>>`

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

fixedPoint :: (Fractional a, Eq a) => (forall s. Forward a s -> Forward a s) -> 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.)

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

`>>>`

0.7390851332151607`last $ take 10 $ fixedPoint cos 1`

extremum :: (Fractional a, Eq a) => (forall s s'. On (Forward (Forward a s') s) -> On (Forward (Forward a s') s)) -> 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.) If the stream
becomes constant (it converges), no further elements are returned.

`>>>`

0.0`last $ take 10 $ extremum cos 1`

# Gradient Ascent/Descent (Reverse AD)

gradientDescent :: (Traversable f, Fractional a, Ord a) => (forall s. Reifies s Tape => f (Reverse a s) -> Reverse a s) -> 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) => (forall s. Reifies s Tape => f (Reverse a s) -> Reverse a s) -> f a -> [f a]Source

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

conjugateGradientDescent :: (Traversable f, Ord a, Fractional a) => (forall t. (Mode t, a ~ Scalar t, Num t) => f t -> t) -> f a -> [f a]Source

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)`

`>>>`

1`rosenbrock [0,0]`

`>>>`

True`rosenbrock (conjugateGradientDescent rosenbrock [0, 0] !! 5) < 0.1`

conjugateGradientAscent :: (Traversable f, Ord a, Fractional a) => (forall t. (Mode t, a ~ Scalar t, Num t) => f t -> t) -> f a -> [f a]Source

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