如何使用 Bash 命令计算 pi
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How can I calculate pi using Bash command
提问by poorvank
I am learning bash scripting. While exploring the math functions i am came across a command which calculated the value of pi.
我正在学习 bash 脚本。在探索数学函数时，我遇到了一个计算 pi 值的命令。
seq -f '4/%g' 1 2 99999 | paste -sd-+ | bc -l
Although i understand how the basic seq command works, I am unable to understand how does the above command works. Can anybody please clarify how does it work.?
虽然我了解基本 seq 命令的工作原理，但我无法理解上述命令的工作原理。任何人都可以请澄清它是如何工作的。？
回答by devnull
This calculates the value of π using Gregory–Leibniz series:
这使用 Gregory-Leibniz 级数计算 π 的值：
seq -f '4/%g' 1 2 99999generates the fractions:
seq -f '4/%g' 1 2 99999生成分数：
4/1
4/3
4/5
4/7
4/9
4/11
4/13
4/15
4/17
4/19
The paste pipeline paste -sd-+combines those with alternate delimiters -and +.
粘贴管道paste -sd-+将那些与备用分隔符-和+.
Finally, bc -lperforms the arithmetic to give the result.
最后，bc -l执行算术运算以给出结果。
EDIT: As noted in the comment, this sequence converges very slowly. Machin's formulahas a significantly higher rate of convergence:
编辑：正如评论中所指出的，这个序列收敛得非常缓慢。 Machin 公式具有明显更高的收敛速度：
Using the same expansion for tan-1(x):
对 tan -1(x)使用相同的展开式：
to compute π, we can see that it produces the correct value to 50 digits1using just the first 50 terms of the series:
为了计算 π，我们可以看到它仅使用系列的前 50 项就可以生成 50 位数字1的正确值：
$ { echo -n "scale=50;"; seq 1 2 100 | xargs -n1 -I{} echo '(16*(1/5)^{}/{}-4*(1/239)^{}/{})';} | paste -sd-+ | bc -l
3.14159265358979323846264338327950288419716939937510
With just 100 terms, the value of π is computed accurately to more than 100 digits:
只需 100 项，π 的值就可以精确计算到 100 多个数字：
$ { echo -n "scale=100;"; seq 1 2 200 | xargs -n1 -I{} echo '(16*(1/5)^{}/{}-4*(1/239)^{}/{})';} | paste -sd-+ | bc -l
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679
1Pi
1圆周率
回答by James Scriven
Not a direct answer to your question about using seq, but pi can be easily computed using bc:
不是您有关 using 问题的直接答案seq，但可以使用pi 轻松计算bc：
echo "scale=1000; 4*a(1)" | bc -l
ais arctan, and this give pi to 1000 digits.
a是反正切，这给 pi 到 1000 位数字。
回答by Nathaniel Flath
seq -f 4 %g 1 2 99999
Gives the data:
给出数据：
4/1
4/3
4/5
...
4/9999
The paste command takes this list and inserts a - between the first two, a + between the second two, etc (and puts it on one line, so):
paste 命令获取此列表并在前两个之间插入 - ，在后两个之间插入 + 等（并将其放在一行上，因此）：
4/1-4/3+4/5-4/7......4/9999
Which is an approximation of pi. The 'bc' program calculates this and prints the value.
这是 pi 的近似值。'bc' 程序计算它并打印该值。
