Numeric.AD.Halley

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.

Halley's Method (Tower AD)

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

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

Examples:

>>> take 10 $ findZero (\x->x^2-4) 1
[1.0,1.8571428571428572,1.9997967892704736,1.9999999999994755,2.0]

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

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

The inverse function inverts a scalar function using Halley'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.

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]

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.)

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

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

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

The extremum function finds an extremum of a scalar function using Halley'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.

>>> take 10 $ extremum cos 1
[1.0,0.29616942658570555,4.59979519460002e-3,1.6220740159042513e-8,0.0]