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Fibonacci: restore matrix power form #1455

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Apr 22, 2025
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Fibonacci: restore matrix power form #1455

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Fibonacci: restore matrix power form
jxu Apr 19, 2025
59c3174
Update src/algebra/fibonacci-numbers.md
adamant-pwn Apr 19, 2025
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15 changes: 14 additions & 1 deletion src/algebra/fibonacci-numbers.md
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Expand Up @@ -157,7 +157,20 @@ F_{n}
\end{pmatrix}
$$

where $F_1 = 1, F_0 = 0$.
where $F_1 = 1, F_0 = 0$.
In fact, since

$$
\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}
= \begin{pmatrix} F_2 & F_1 \\ F_1 & F_0 \end{pmatrix}
$$

we can use the matrix directly:

$$
\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n
= \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}
$$

Thus, in order to find $F_n$ in $O(\log n)$ time, we must raise the matrix to n. (See [Binary exponentiation](binary-exp.md))

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