# Fastest Fibonacci Sequence/Number Computation

The fastest way (in O(n)) of calculating Fibonacci sequence is by using matrix multiplication approach using following relation.
$\begin{bmatrix}0 & 1 \\ 1 & 1 \end{bmatrix}^n = \begin{bmatrix} F_{n – 1} & F_n \\ F_n & F_{n + 1}\end{bmatrix}$
Calculating $F_{34}$ is, therefore, multiplying the matrix $\begin{bmatrix}0 & 1 \\ 1 & 1\end{bmatrix}$ 34 times. The $a_{01}$ or $a_{10}$ gives the right fibonacci number. In fact $F_{34}$ can be calculated in less than 34 multiplication in following away.
$\begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^2 = \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix} X \ \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix} \\ \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^4 = \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^2 X \ \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^2 \\ \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^8 = \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^4 X \ \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^4 \\ \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^{16} = \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^8 X \ \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^8 \\ \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^{32} = \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^{16} X \ \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^{16} \\ \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^{34} = \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^{32} X \ \begin{bmatrix} 0 & 1 \\ 1 & 1\end{bmatrix}^{2} \\$

This means, the multiplication can be done in logarithmic time. The C code for this computation is given below (also available on GitHub).

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