Safe Haskell  None 

Language  Haskell2010 
The Converge type class.
Documentation
class Converge a where Source #
If a type is an instance of Converge then it represents a stream of values which are increasingly accurate approximations of a desired value
converge :: a > Maybe (Element a) Source #
converge
is a function that returns the value the stream is converging
to. If given a stream which doens't converge to a single value then
converge
will not terminate.
If the stream is empty then it should return nothing.
>>>
let initialGuess = 1 :: Double
>>>
let improve x = (x + 121 / x) / 2
>>>
converge (iterate improve initialGuess)
Just 11.0
>>>
converge [] :: Maybe [Int]
Nothing
convergeErr :: Ord (Element a) => (Element a > Element a) > a > Maybe (Element a) Source #
convergeErr
is a function that returns the value the stream is
converging to. It also takes a function err which returns a value which
varies monotonically with the error of the value in the stream. This can
be used to ensure that when convergeErr
terminates when given a
nonconverging stream or a stream which enters a cycle close to the
solution. See the documentation for the CReal instance for a caveat with
that implementation.
It's often the case that streams generated with approximation functions such as Newton's method will generate worse approximations for some number of steps until they find the "zone of convergence". For these cases it's necessary to drop some values of the stream before handing it to convergeErr.
For example trying to find the root of the following funciton f
with a
poor choice of starting point. Although this doesn't find the root, it
doesn't fail to terminate.
>>>
let f x = x ^ 3  2 * x + 2
>>>
let f' x = 3 * x ^ 2  2
>>>
let initialGuess = 0.1 :: Float
>>>
let improve x = x  f x / f' x
>>>
let err x = abs (f x)
>>>
convergeErr err (iterate improve initialGuess)
Just 1.0142132
Instances
Eq a => Converge [a] Source #  Every list of equatable values is an instance of 
Converge [CReal n] Source #  The overlapping instance for It's important to note when the error function reaches zero this function
behaves like Find where log x = π using Newton's method
