Documentation

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 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 non-converging 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