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. AD s (Tower a) -> AD s (Tower 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]

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

findZeroNoEq :: Fractional a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> [a]

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) => (forall s. AD s (Tower a) -> AD s (Tower 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!

inverseNoEq :: Fractional a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> a -> [a]

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) => (forall s. AD s (Tower a) -> AD s (Tower 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

fixedPointNoEq :: Fractional a => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> [a]

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) => (forall s. AD s (On (Forward (Tower a))) -> AD s (On (Forward (Tower 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]

extremumNoEq :: Fractional a => (forall s. AD s (On (Forward (Tower a))) -> AD s (On (Forward (Tower a)))) -> a -> [a]

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.