Fixed point combinator in Haskell

Question

The fixed point combinator doesn\'t always produce the right answer given the definition:

fix f = f (fix f)

The following code does not ter

Answer 1

You probably meant iterate:

*Main> take 8 $ iterate (^2) (0.0 ::Float)
[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0]
*Main> take 8 $ iterate (^2) (0.001 ::Float)
[1.0e-3,1.0000001e-6,1.0000002e-12,1.0000004e-24,0.0,0.0,0.0,0.0]

*Main> take 8 $ iterate (^2) (0.999 ::Float)
[0.999,0.99800104,0.9960061,0.9920281,0.9841198,0.96849173,0.93797624,0.8797994]
*Main> take 8 $ iterate (^2) (1.0 ::Float)
[1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0]
*Main> take 8 $ iterate (^2) (1.001 ::Float)
[1.001,1.002001,1.0040061,1.0080284,1.0161213,1.0325024,1.0660613,1.1364866]

Here you have all the execution history explicitly available for your analysis. You can attempt to detect the fixed point with

fixed f from = snd . head 
                   . until ((< 1e-16).abs.uncurry (-).head) tail 
               $ _S zip tail history
  where history = iterate f from
        _S f g x = f x (g x)

and then

*Main> fixed (^2) (0.999 :: Float)
0.0

but trying fixed (^2) (1.001 :: Float) will loop indefinitely, so you'd need to develop separate testing for convergence, and even then detection of repellent fixed points like 1.0 will need more elaborate investigation.