Can fixed-point functions be used on polynomials?

Question

I was thinking of a way to represent algebraic numbers in Haskell as a stream of approximations. You could probably do this by some root finding algorithm. But that's no fun. So you could add x to the polynomial, reducing the problem to finding it's fixed points.

So if you have a function in Haskell like

f :: Double -> Double
f x = x ^ 2 + x

I don't conceptually understand why fix doesn't work, which is to say, I can easily verify for myself that it doesn't work, but isn't 0 the true least fixed point of f? Is there another simple (as in definition size) fixed point function that would work?

Answer 1

Here is the implementation of the fix function:

fix :: (a -> a) -> a
fix f = let x = f x in x

It doesn't work for primitive types like Double. It's intended for types that have a more complex structure to them. For instance:

g :: Maybe Int -> Maybe Int
g i = Just $ case i of
    Nothing -> 3
    Just _ -> 4

This function will work with fix because it yields information about its result faster than it reads its input. in other words, the Just portion is known without looking at i at all, which enables it to reach a fixed point.

When your function is Double -> Double, and examines its input, fix won't work because there's no way to partially evaluate a Double.