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

Stability | experimental |

Maintainer | ekmett@gmail.com |

Safe Haskell | Safe-Infered |

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, Eq a) => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]
- inverse :: (Fractional a, Eq a) => (forall s. Mode s => AD s a -> AD s a) -> a -> a -> [a]
- fixedPoint :: (Fractional a, Eq a) => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]
- extremum :: (Fractional a, Eq a) => (forall s. Mode s => AD s a -> AD s a) -> a -> [a]

# Halley's Method (Tower AD)

findZero :: (Fractional a, Eq a) => (forall s. Mode s => AD s a -> AD s 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, Eq a) => (forall s. Mode s => AD s a -> AD s 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, Eq a) => (forall s. Mode s => AD s a -> AD s 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