The Fibonacci sequence
From HaskellWiki
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| - | Implementing the [http://en.wikipedia.org/wiki/Fibonacci_number | + | Implementing the [http://en.wikipedia.org/wiki/Fibonacci_number Fibonacci sequence] is considered the "Hello, world!" of Haskell programming. This page collects Haskell implementations of the sequence. |
| - | == Naive | + | == Naive definition == |
<haskell> | <haskell> | ||
| Line 9: | Line 9: | ||
</haskell> | </haskell> | ||
| - | == | + | == Linear-time implementations == |
| + | One can compute the first ''n'' Fibonacci numbers with ''O(n)'' additions. | ||
| + | If <hask>fibs</hask> is the infinite list of Fibonacci numbers, one can define | ||
<haskell> | <haskell> | ||
| - | fib = | + | fib n = fibs!!n |
</haskell> | </haskell> | ||
| - | == | + | === Canonical zipWith implementation === |
<haskell> | <haskell> | ||
| - | + | fibs = 0 : 1 : zipWith (+) fibs (tail fibs) | |
</haskell> | </haskell> | ||
| - | == With | + | === With scanl === |
<haskell> | <haskell> | ||
| - | + | fibs = fix ((0:) . scanl (+) 1) | |
</haskell> | </haskell> | ||
| - | == A fairly fast version, using some identities == | + | === With unfoldr === |
| + | |||
| + | <haskell> | ||
| + | fibs = unfoldr (\(f1,f2) -> Just (f1,(f2,f1+f2))) (0,1) | ||
| + | </haskell> | ||
| + | |||
| + | == Log-time implementations == | ||
| + | |||
| + | === Using 2x2 matrices === | ||
| + | |||
| + | Using [[Prelude_extensions#Matrices|simple matrices]], | ||
| + | <haskell> | ||
| + | fib n = head (apply (Matrix [[1,1], [1,0]] ^ n) [0,1]) | ||
| + | </haskell> | ||
| + | |||
| + | === A fairly fast version, using some identities === | ||
<haskell> | <haskell> | ||
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</haskell> | </haskell> | ||
| - | == Another fast fib == | + | === Another fast fib === |
<haskell> | <haskell> | ||
| Line 55: | Line 72: | ||
</haskell> | </haskell> | ||
| - | == Fastest Fib in the West == | + | === Fastest Fib in the West === |
This was contributed by [http://www.haskell.org/pipermail/haskell-cafe/2005-January/008839.html wli] | This was contributed by [http://www.haskell.org/pipermail/haskell-cafe/2005-January/008839.html wli] | ||
| + | (It assumes that the sequence starts with 1.) | ||
<haskell> | <haskell> | ||
| - | |||
import Data.List | import Data.List | ||
| - | + | fib1 n = snd . foldl fib' (1, 0) . map (toEnum . fromIntegral) $ unfoldl divs n | |
where | where | ||
unfoldl f x = case f x of | unfoldl f x = case f x of | ||
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| p = (f*(f+2*g), f^2 + g^2) | | p = (f*(f+2*g), f^2 + g^2) | ||
| otherwise = (f^2+g^2, g*(2*f-g)) | | otherwise = (f^2+g^2, g*(2*f-g)) | ||
| - | |||
| - | |||
</haskell> | </haskell> | ||
Revision as of 23:02, 9 May 2007
Implementing the Fibonacci sequence is considered the "Hello, world!" of Haskell programming. This page collects Haskell implementations of the sequence.
Contents |
1 Naive definition
fib 0 = 0 fib 1 = 1 fib n = fib (n-1) + fib (n-2)
2 Linear-time implementations
One can compute the first n Fibonacci numbers with O(n) additions.
Iffibs
fib n = fibs!!n
2.1 Canonical zipWith implementation
fibs = 0 : 1 : zipWith (+) fibs (tail fibs)
2.2 With scanl
fibs = fix ((0:) . scanl (+) 1)
2.3 With unfoldr
fibs = unfoldr (\(f1,f2) -> Just (f1,(f2,f1+f2))) (0,1)
3 Log-time implementations
3.1 Using 2x2 matrices
Using simple matrices,
fib n = head (apply (Matrix [[1,1], [1,0]] ^ n) [0,1])
3.2 A fairly fast version, using some identities
fib 0 = 0 fib 1 = 1 fib n | even n = f1 * (f1 + 2 * f2) | n `mod` 4 == 1 = (2 * f1 + f2) * (2 * f1 - f2) + 2 | otherwise = (2 * f1 + f2) * (2 * f1 - f2) - 2 where k = n `div` 2 f1 = fib k f2 = fib (k-1)
3.3 Another fast fib
fib = fst . fib2 -- | Return (fib n, fib (n + 1)) fib2 0 = (1, 1) fib2 1 = (1, 2) fib2 n | even n = (a*a + b*b, c*c - a*a) | otherwise = (c*c - a*a, b*b + c*c) where (a,b) = fib2 (n `div` 2 - 1) c = a + b
3.4 Fastest Fib in the West
This was contributed by wli (It assumes that the sequence starts with 1.)
import Data.List fib1 n = snd . foldl fib' (1, 0) . map (toEnum . fromIntegral) $ unfoldl divs n where unfoldl f x = case f x of Nothing -> [] Just (u, v) -> unfoldl f v ++ [u] divs 0 = Nothing divs k = Just (uncurry (flip (,)) (k `divMod` 2)) fib' (f, g) p | p = (f*(f+2*g), f^2 + g^2) | otherwise = (f^2+g^2, g*(2*f-g))
