Math.Sequence.Converge

Documentation

converge :: Eq a => [a] -> a

Take items from the list until two successive items are equal and return the second of them (or an item is not equal to itself, to handle NaN without a RealFloat context. In this case, the first item of the pair is returned) . If the list ends before a match is found, returns the last element of the list.

convergeTo :: (Fractional a, Ord a) => a -> a -> [a] -> a

convergeTo absEps relEps xs takes items from xs until two successive items x and y are within either absEps or relEps * max (abs x) (abs y) of each other, in which case the second of the pair is returned, or until an item is found that does not equal itself (which would typically be a NaN), in which case the preceding item is returned. If the list ends before a match is found, the last element of the list is returned.

For example, approximating the golden mean by applying Newton's method to find a root of x^2 - x - 1:

 phi :: Rational
 phi = convergeTo 1e-100 0 (iterate (\x -> (x*x + 1) / (2*x-1)) 1)

convergeBy :: ([a] -> Maybe b) -> (a -> Maybe b) -> [a] -> Maybe b

convergeBy f end xs looks through xs for the first segment for which f returns a value, and returns that value. Typically f would be something like:

 f (a:b:_)
    | abs(a-b) <= eps
    = Just (0.5 * (a + b))
 f _ = Nothing

If no such segment is found, applies end to the last item in the list and returns the result. If the list was empty, returns Nothing.