| Copyright | (c) Edward Kmett 2010-2014 |
|---|---|
| License | BSD3 |
| Maintainer | ekmett@gmail.com |
| Stability | experimental |
| Portability | GHC only |
| Safe Haskell | None |
| Language | Haskell2010 |
Numeric.AD.Halley
Contents
Description
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. AD s (Tower a) -> AD s (Tower a)) -> a -> [a]
- inverse :: (Fractional a, Eq a) => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> a -> [a]
- fixedPoint :: (Fractional a, Eq a) => (forall s. AD s (Tower a) -> AD s (Tower a)) -> a -> [a]
- extremum :: (Fractional a, Eq a) => (forall s. AD s (On (Forward (Tower a))) -> AD s (On (Forward (Tower a)))) -> a -> [a]
Halley's Method (Tower AD)
findZero :: (Fractional a, Eq a) => (forall s. AD s (Tower a) -> AD s (Tower 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.) 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
inverse :: (Fractional a, Eq a) => (forall s. AD s (Tower a) -> AD s (Tower 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.) 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. AD s (Tower a) -> AD s (Tower 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.)
If the stream becomes constant ("it converges"), no further elements are returned.
>>>last $ take 10 $ fixedPoint cos 10.7390851332151607
extremum :: (Fractional a, Eq a) => (forall s. AD s (On (Forward (Tower a))) -> AD s (On (Forward (Tower a)))) -> a -> [a] Source
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]