From 115cc22af2ba00e76f123f2a382afc44ff9710d4 Mon Sep 17 00:00:00 2001 From: jxu <7989982+jxu@users.noreply.github.com> Date: Fri, 18 Apr 2025 23:11:59 -0400 Subject: [PATCH 1/2] Fibonacci: restore matrix power form Maybe a dotted line would show the matrix [[F2, F1],[F1,F0]] can be viewed as two column vectors. Using only the matrix power saves one matrix-vector multiply. --- src/algebra/fibonacci-numbers.md | 14 +++++++++++++- 1 file changed, 13 insertions(+), 1 deletion(-) diff --git a/src/algebra/fibonacci-numbers.md b/src/algebra/fibonacci-numbers.md index e63e52081..e1acfdcb0 100644 --- a/src/algebra/fibonacci-numbers.md +++ b/src/algebra/fibonacci-numbers.md @@ -157,7 +157,19 @@ 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} +$$ Thus, in order to find $F_n$ in $O(\log n)$ time, we must raise the matrix to n. (See [Binary exponentiation](binary-exp.md)) From 59c31747c57735cfe83639f9a1dac64bb800dd78 Mon Sep 17 00:00:00 2001 From: Oleksandr Kulkov Date: Sat, 19 Apr 2025 09:56:55 +0200 Subject: [PATCH 2/2] Update src/algebra/fibonacci-numbers.md --- src/algebra/fibonacci-numbers.md | 1 + 1 file changed, 1 insertion(+) diff --git a/src/algebra/fibonacci-numbers.md b/src/algebra/fibonacci-numbers.md index e1acfdcb0..dd8e1bd1a 100644 --- a/src/algebra/fibonacci-numbers.md +++ b/src/algebra/fibonacci-numbers.md @@ -159,6 +159,7 @@ $$ 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} pFad - Phonifier reborn

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