https://github.com/adrian-lin-1-0-0/fibonacci
https://github.com/adrian-lin-1-0-0/fibonacci
Last synced: 3 months ago
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- Host: GitHub
- URL: https://github.com/adrian-lin-1-0-0/fibonacci
- Owner: adrian-lin-1-0-0
- Created: 2023-04-14T10:56:14.000Z (about 2 years ago)
- Default Branch: master
- Last Pushed: 2023-04-14T11:04:05.000Z (about 2 years ago)
- Last Synced: 2025-01-25T18:43:41.514Z (5 months ago)
- Language: Go
- Size: 4.88 KB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# Fibonacci
## Q-Matrix
$$\begin{aligned}
Q &= \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} F_2 & F_1 \\ F_1 & F_0 \end{bmatrix}
\end{aligned}$$
$$\begin{aligned}
Q^{n+1} &= QQ^n \\
&= \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{bmatrix} \\
&= \begin{bmatrix} F_{n+1} + F_n & F_n + F_{n-1} \\ F_{n+1} &F_n \end{bmatrix} \\
&= \begin{bmatrix} F_{n+2} & F_{n+1} \\ F_{n+1} & F_{n} \end{bmatrix}
\end{aligned}$$
## Fast doubling
$$\begin{aligned}
\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^n = \begin{bmatrix} F(n+1) & F(n) \\ F(n) & F(n-1) \end{bmatrix}
\end{aligned}$$
$$\begin{aligned}
\begin{bmatrix} F(2n+1) \\ F(2n) \end{bmatrix}^n &= \begin{bmatrix} F(n+1) & F(n) \\ F(n) & F(n-1) \end{bmatrix}^{2n} \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\
&= \begin{bmatrix} F(n+1) & F(n) \\ F(n) & F(n-1) \end{bmatrix} \begin{bmatrix} F(n+1) & F(n) \\ F(n) & F(n-1) \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\
&= \begin{bmatrix} F(n+1)^2 + F(n)^2 \\ F(n)F(n+1) + F(n-1)F(n) \end{bmatrix}
\end{aligned}$$
$$\begin{aligned}
F(2n+1) &= F(n+1)^2 + F(n)^2 \\
\end{aligned}$$
$$\begin{aligned}
F(2n) &= F(n)F(n+1) + F(n-1)F(n) \\
&= F(n)[F(n+1)+F(n-1) \\
&= F(n)[F(n+1)+F(n-1)+F(n)-F(n)] \\
&= F(n)[F(n+1)+F(n+1)-F(n)] \\
&= F(n)[2*F(n+1)-F(n)]
\end{aligned}$$
Golang code
```go
func fastDoubling(n int) int {
if n == 0 {
return 0
}
fn := 1 //F(n)
fn1 := 1 //F(n+1)
f2n := 1 //F(2n)
f2n1 := 0 //F(2n+1)
i := 1
for i < n {
if (i << 1) <= n {
f2n1, f2n = fn1*fn1+fn*fn, fn*(2*fn1-fn)
fn, fn1 = f2n, f2n1
i = i << 1
} else {
fn, f2n = f2n, f2n1
f2n1 = fn + f2n1 // F(2n) + F(2n+1)
i++
}
}
return f2n
}
```
```
BenchmarkFastDoubling-8 1000000000 0.0000018 ns/op 0 B/op 0 allocs/op
BenchmarkRecursionChannel-8 1000000000 0.002087 ns/op 0 B/op 0 allocs/op
BenchmarkTailCall-8 1000000000 0.0003478 ns/op 0 B/op 0 allocs/op
BenchmarkForLoop-8 1000000000 0.0000029 ns/op 0 B/op 0 allocs/op
```