# Difference between revisions of "The Fibonacci sequence"

RossPaterson (talk | contribs) m (Another fast fib assumes that the sequence starts with 1) |
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== Linear operation implementations == | == Linear operation implementations == | ||

− | === | + | === With state === |

+ | Haskell translation of python algo | ||

<haskell> | <haskell> | ||

− | fib 0 = | + | {- def fib(n): |

− | + | a, b = 0, 1 | |

− | fib n | + | for _ in xrange(n): |

− | + | a, b = b, a + b | |

− | + | return a -} | |

− | + | </haskell> | |

− | + | ||

− | + | ==== Tail recursive ==== | |

+ | |||

+ | Using accumulator argument for state passing | ||

+ | <haskell> | ||

+ | {-# LANGUAGE BangPatterns #-} | ||

+ | fib n = go n (0,1) | ||

+ | where | ||

+ | go !n (!a, !b) | n==0 = a | ||

+ | | otherwise = go (n-1) (b, a+b) | ||

+ | </haskell> | ||

+ | |||

+ | ==== Monadic ==== | ||

+ | <haskell> | ||

+ | import Control.Monad.State | ||

+ | fib n = flip evalState (0,1) $ do | ||

+ | forM [0..(n-1)] $ \_ -> do | ||

+ | (a,b) <- get | ||

+ | put (b,a+b) | ||

+ | (a,b) <- get | ||

+ | return a | ||

</haskell> | </haskell> | ||

Line 38: | Line 58: | ||

<haskell> | <haskell> | ||

fibs = 0 : 1 : zipWith (+) fibs (tail fibs) | fibs = 0 : 1 : zipWith (+) fibs (tail fibs) | ||

+ | </haskell> | ||

+ | |||

+ | ==== With direct self-reference ==== | ||

+ | |||

+ | <haskell> | ||

+ | fibs = 0 : 1 : next fibs | ||

+ | where | ||

+ | next (a : t@(b:_)) = (a+b) : next t | ||

</haskell> | </haskell> | ||

Line 44: | Line 72: | ||

<haskell> | <haskell> | ||

fibs = scanl (+) 0 (1:fibs) | fibs = scanl (+) 0 (1:fibs) | ||

− | fibs = 0 : scanl (+) | + | fibs = 0 : scanl (+) 1 fibs |

</haskell> | </haskell> | ||

The recursion can be replaced with <hask>fix</hask>: | The recursion can be replaced with <hask>fix</hask>: | ||

Line 51: | Line 79: | ||

fibs = fix ((0:) . scanl (+) 1) | fibs = fix ((0:) . scanl (+) 1) | ||

</haskell> | </haskell> | ||

+ | |||

+ | The <code>fix</code> used here has to be implemented through sharing, <code>fix f = xs where xs = f xs</code>, not code replication, <code>fix f = f (fix f)</code>, to avoid quadratic behaviour. | ||

==== With unfoldr ==== | ==== With unfoldr ==== | ||

Line 63: | Line 93: | ||

fibs = map fst $ iterate (\(a,b) -> (b,a+b)) (0,1) | fibs = map fst $ iterate (\(a,b) -> (b,a+b)) (0,1) | ||

</haskell> | </haskell> | ||

+ | |||

+ | === A version using some identities === | ||

+ | |||

+ | <haskell> | ||

+ | 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) | ||

+ | </haskell> | ||

+ | |||

+ | This seems to use <math>O(log^2 n)</math> calls to <code>fib</code>. | ||

== Logarithmic operation implementations == | == Logarithmic operation implementations == | ||

Line 68: | Line 113: | ||

=== Using 2x2 matrices === | === Using 2x2 matrices === | ||

− | The argument of <hask>iterate</hask> above is a [http://en.wikipedia.org/wiki/Linear_map linear transformation], | + | The argument of <hask>iterate</hask> above is a [http://en.wikipedia.org/wiki/Linear_map linear transformation], so we can represent it as matrix and compute the ''n''th power of this matrix with ''O(log n)'' multiplications and additions. |

− | so we can represent it as matrix and compute the ''n''th power of this matrix with ''O(log n)'' multiplications and additions. | ||

For example, using the [[Prelude_extensions#Matrices|simple matrix implementation]] in [[Prelude extensions]], | For example, using the [[Prelude_extensions#Matrices|simple matrix implementation]] in [[Prelude extensions]], | ||

<haskell> | <haskell> | ||

Line 100: | Line 144: | ||

import Data.List | import Data.List | ||

− | fib1 n = snd . foldl | + | 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 | ||

Line 109: | Line 153: | ||

divs k = Just (uncurry (flip (,)) (k `divMod` 2)) | divs k = Just (uncurry (flip (,)) (k `divMod` 2)) | ||

− | + | fib_ (f, g) p | |

| 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)) | ||

Line 120: | Line 164: | ||

fib :: Int -> Integer | fib :: Int -> Integer | ||

− | fib n = snd . | + | fib n = snd . foldl_ fib_ (1, 0) . dropWhile not $ |

[testBit n k | k <- let s = bitSize n in [s-1,s-2..0]] | [testBit n k | k <- let s = bitSize n in [s-1,s-2..0]] | ||

where | where | ||

− | + | fib_ (f, g) p | |

| p = (f*(f+2*g), ss) | | p = (f*(f+2*g), ss) | ||

| otherwise = (ss, g*(2*f-g)) | | otherwise = (ss, g*(2*f-g)) | ||

where ss = f*f+g*g | where ss = f*f+g*g | ||

+ | foldl_ = foldl' -- ' | ||

</haskell> | </haskell> | ||

Line 146: | Line 191: | ||

phi = (1 + sq5) / 2 | phi = (1 + sq5) / 2 | ||

</haskell> | </haskell> | ||

+ | |||

+ | == Generalization of Fibonacci numbers == | ||

+ | The numbers of the traditional Fibonacci sequence are formed by summing its two preceding numbers, with starting values 0 and 1. Variations of the sequence can be obtained by using different starting values and summing a different number of predecessors. | ||

+ | |||

+ | === Fibonacci ''n''-Step Numbers === | ||

+ | The sequence of Fibonacci ''n''-step numbers are formed by summing ''n'' predecessors, using (''n''-1) zeros and a single 1 as starting values: | ||

+ | <math> | ||

+ | F^{(n)}_k := | ||

+ | \begin{cases} | ||

+ | 0 & \mbox{if } 0 \leq k < n-1 \\ | ||

+ | 1 & \mbox{if } k = n-1 \\ | ||

+ | \sum\limits_{i=k-n}^{(k-1)} F^{(n)}_i & \mbox{otherwise} \\ | ||

+ | \end{cases} | ||

+ | </math> | ||

+ | |||

+ | Note that the summation in the current definition has a time complexity of ''O(n)'', assuming we memoize previously computed numbers of the sequence. We can do better than. Observe that in the following Tribonacci sequence, we compute the number 81 by summing up 13, 24 and 44: | ||

+ | |||

+ | <math> | ||

+ | F^{(3)} = 0,0,1,1,2,4,7,\underbrace{13,24,44},81,149, \ldots | ||

+ | </math> | ||

+ | |||

+ | The number 149 is computed in a similar way, but can also be computed as follows: | ||

+ | |||

+ | <math> | ||

+ | 149 = 24 + 44 + 81 = (13 + 24 + 44) + 81 - 13 = 81 + 81 - 13 = 2\cdot 81 - 13 | ||

+ | </math> | ||

+ | |||

+ | And hence, an equivalent definition of the Fibonacci ''n''-step numbers sequence is: | ||

+ | |||

+ | <math> | ||

+ | F^{(n)}_k := | ||

+ | \begin{cases} | ||

+ | 0 & \mbox{if } 0 \leq k < n-1 \\ | ||

+ | 1 & \mbox{if } k = n-1 \\ | ||

+ | 1 & \mbox{if } k = n \\ | ||

+ | F^{(n)}_k := 2F^{(n)}_{k-1}-F^{(n)}_{k-(n+1)} & \mbox{otherwise} \\ | ||

+ | \end{cases} | ||

+ | </math> | ||

+ | |||

+ | ''(Notice the extra case that is needed)'' | ||

+ | |||

+ | Transforming this directly into Haskell gives us: | ||

+ | <haskell> | ||

+ | nfibs n = replicate (n-1) 0 ++ 1 : 1 : | ||

+ | zipWith (\b a -> 2*b-a) (drop n (nfibs n)) (nfibs n) | ||

+ | </haskell> | ||

+ | This version, however, is slow since the computation of <hask>nfibs n</hask> is not shared. Naming the result using a let-binding and making the lambda pointfree results in: | ||

+ | <haskell> | ||

+ | nfibs n = let r = replicate (n-1) 0 ++ 1 : 1 : zipWith ((-).(2*)) (drop n r) r | ||

+ | in r | ||

+ | </haskell> | ||

+ | |||

== See also == | == See also == |

## Latest revision as of 17:37, 5 November 2016

Implementing the Fibonacci sequence is considered the "Hello, world!" of Haskell programming. This page collects Haskell implementations of the sequence.

## Contents

## Naive definition

The standard definition can be expressed directly:

```
fib 0 = 0
fib 1 = 1
fib n = fib (n-1) + fib (n-2)
```

This implementation requires *O(fib n)* additions.

## Linear operation implementations

### With state

Haskell translation of python algo

```
{- def fib(n):
a, b = 0, 1
for _ in xrange(n):
a, b = b, a + b
return a -}
```

#### Tail recursive

Using accumulator argument for state passing

```
{-# LANGUAGE BangPatterns #-}
fib n = go n (0,1)
where
go !n (!a, !b) | n==0 = a
| otherwise = go (n-1) (b, a+b)
```

#### Monadic

```
import Control.Monad.State
fib n = flip evalState (0,1) $ do
forM [0..(n-1)] $ \_ -> do
(a,b) <- get
put (b,a+b)
(a,b) <- get
return a
```

### Using the infinite list of Fibonacci numbers

One can compute the first *n* Fibonacci numbers with *O(n)* additions.
If `fibs`

is the infinite list of Fibonacci numbers, one can define

```
fib n = fibs!!n
```

#### Canonical zipWith implementation

```
fibs = 0 : 1 : zipWith (+) fibs (tail fibs)
```

#### With direct self-reference

```
fibs = 0 : 1 : next fibs
where
next (a : t@(b:_)) = (a+b) : next t
```

#### With scanl

```
fibs = scanl (+) 0 (1:fibs)
fibs = 0 : scanl (+) 1 fibs
```

The recursion can be replaced with `fix`

:

```
fibs = fix (scanl (+) 0 . (1:))
fibs = fix ((0:) . scanl (+) 1)
```

The `fix`

used here has to be implemented through sharing, `fix f = xs where xs = f xs`

, not code replication, `fix f = f (fix f)`

, to avoid quadratic behaviour.

#### With unfoldr

```
fibs = unfoldr (\(a,b) -> Just (a,(b,a+b))) (0,1)
```

#### With iterate

```
fibs = map fst $ iterate (\(a,b) -> (b,a+b)) (0,1)
```

### A 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)
```

This seems to use calls to `fib`

.

## Logarithmic operation implementations

### Using 2x2 matrices

The argument of `iterate`

above is a linear transformation, so we can represent it as matrix and compute the *n*th power of this matrix with *O(log n)* multiplications and additions.
For example, using the simple matrix implementation in Prelude extensions,

```
fib n = head (apply (Matrix [[0,1], [1,1]] ^ n) [0,1])
```

This technique works for any linear recurrence.

### Another fast fib

(Assumes that the sequence starts with 1.)

```
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
```

### 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))
```

An even faster version, given later by wli on the IRC channel.

```
import Data.List
import Data.Bits
fib :: Int -> Integer
fib n = snd . foldl_ fib_ (1, 0) . dropWhile not $
[testBit n k | k <- let s = bitSize n in [s-1,s-2..0]]
where
fib_ (f, g) p
| p = (f*(f+2*g), ss)
| otherwise = (ss, g*(2*f-g))
where ss = f*f+g*g
foldl_ = foldl' -- '
```

## Constant-time implementations

The Fibonacci numbers can be computed in constant time using Binet's formula. However, that only works well within the range of floating-point numbers available on your platform. Implementing Binet's formula in such a way that it computes exact results for all integers generally doesn't result in a terribly efficient implementation when compared to the programs above which use a logarithmic number of operations (and work in linear time).

Beyond that, you can use unlimited-precision floating-point numbers, but the result will probably not be any better than the log-time implementations above.

### Using Binet's formula

```
fib n = round $ phi ** fromIntegral n / sq5
where
sq5 = sqrt 5 :: Double
phi = (1 + sq5) / 2
```

## Generalization of Fibonacci numbers

The numbers of the traditional Fibonacci sequence are formed by summing its two preceding numbers, with starting values 0 and 1. Variations of the sequence can be obtained by using different starting values and summing a different number of predecessors.

### Fibonacci *n*-Step Numbers

The sequence of Fibonacci *n*-step numbers are formed by summing *n* predecessors, using (*n*-1) zeros and a single 1 as starting values:

Note that the summation in the current definition has a time complexity of *O(n)*, assuming we memoize previously computed numbers of the sequence. We can do better than. Observe that in the following Tribonacci sequence, we compute the number 81 by summing up 13, 24 and 44:

The number 149 is computed in a similar way, but can also be computed as follows:

And hence, an equivalent definition of the Fibonacci *n*-step numbers sequence is:

*(Notice the extra case that is needed)*

Transforming this directly into Haskell gives us:

```
nfibs n = replicate (n-1) 0 ++ 1 : 1 :
zipWith (\b a -> 2*b-a) (drop n (nfibs n)) (nfibs n)
```

This version, however, is slow since the computation of `nfibs n`

is not shared. Naming the result using a let-binding and making the lambda pointfree results in:

```
nfibs n = let r = replicate (n-1) 0 ++ 1 : 1 : zipWith ((-).(2*)) (drop n r) r
in r
```

## See also

- Naive parallel, multicore version
- Fibonacci primes in parallel
- Discussion at haskell cafe
- Some other nice solutions
- In Project Euler, some of the problems involve Fibonacci numbers. There are some solutions in Haskell (
**Spoiler Warning:**Do not look at solutions to Project Euler problems until you have solved the problems on your own.):