TITLE: Fastest known way to computing the $n$-th Fibonacci number?
QUESTION [6 upvotes]: Using the recursive formula and a bit of memory, you can compute this number in linear time. 
But there is also the direct formula. Unfortunately, it used irrational numbers, so first need to fix an adequate ring in which you do the computations (like the rationals and $\sqrt5$). Then you need to compute and subtract the powers of the irrational numbers. So it is not obvious how many steps you really need.
Or is there an even faster way?

REPLY [3 votes]: The formula $$F_{2n+1}=F_{n+1}^2+F_n^2\\F_{2n}=F_n(2F_{n+1}-F_n)$$ is also $\log(n) $ this identity follows by using the matrix Misha wrote and writing $A^{2n}=A^nA^n$