Copyright | (c) Edward Kmett 2010-2021 |
---|---|

License | BSD3 |

Maintainer | ekmett@gmail.com |

Stability | experimental |

Portability | GHC only |

Safe Haskell | Safe-Inferred |

Language | Haskell2010 |

## Synopsis

- findZero :: (Fractional a, Eq a) => (Forward a -> Forward a) -> a -> [a]
- findZeroNoEq :: Fractional a => (Forward a -> Forward a) -> a -> [a]
- inverse :: (Fractional a, Eq a) => (Forward a -> Forward a) -> a -> a -> [a]
- inverseNoEq :: Fractional a => (Forward a -> Forward a) -> a -> a -> [a]
- fixedPoint :: (Fractional a, Eq a) => (Forward a -> Forward a) -> a -> [a]
- fixedPointNoEq :: Fractional a => (Forward a -> Forward a) -> a -> [a]
- extremum :: (Fractional a, Eq a) => (On (Forward (Forward a)) -> On (Forward (Forward a))) -> a -> [a]
- extremumNoEq :: Fractional a => (On (Forward (Forward a)) -> On (Forward (Forward a))) -> a -> [a]
- gradientDescent :: (Traversable f, Fractional a, Ord a) => (f (Kahn a) -> Kahn a) -> f a -> [f a]
- gradientAscent :: (Traversable f, Fractional a, Ord a) => (f (Kahn a) -> Kahn a) -> f a -> [f a]

# Newton's Method (Forward)

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

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

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] 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)`

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

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] 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`

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

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] 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`

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

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

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