Questions tagged [fibonacci]

The Fibonacci sequence is the sequence defined by

F(0) = 0
F(1) = 1
F(n + 2) = F(n) + F(n + 1).  

The first few terms are 0, 1, 1, 2, 3, 5, and 8.

The most efficient way to compute the first N values is to iterate over an array using the above formula, resulting in O(N) operations (O(N²) if digit or bit operations are counted). The recursive implementation should be generally avoided, since it is O(φN) where φ is the golden ratio and is equal to (1+sqrt(5))/2. However, by using a cache for already computed values, it can be as fast as the iterative implementation.

One efficient method for computing single Fibonacci numbers is

Fib(n) = Round(  Power( 0.5*(1+Sqrt(5)), n ) / Sqrt(5)  );

The square root and power have to be computed in sufficient precision, roughly, Fib(n) requires about 0.2*n decimal digits or about 0.7*n bits in the integer result.

Another method is based on the fact that the matrix

( Fib[n+1] Fib[ n ] )
( Fib[ n ] Fib[n-1] )

is the n-th power of the matrix

( 1 1 ) which is ( Fib[2] Fib[1] )
( 1 0 ) equal to ( Fib[1] Fib[0] )

This is the basis of a halving-and-squaring method that computes Fib[N] in O(log(N)) operations, completely in (big) integer arithmetic.

If one accounts for the complexity of big integer multiplication, the complexity raises to O( M(N) ) digit or bit operations, where M(d) is the complexity of the multiplication of numbers of bit length d. Typical methods have M(d)=O(d*log(d)) for FFT based multiplication, M(d)=O(d^w) with w=log₂(3) for Karatsuba and smaller w for the Toom-Cook methods.