数学研发论坛

 找回密码
 欢迎注册
查看: 184|回复: 7

[转载] 计算圆周率的Chudnovsky公式

[复制链接]
发表于 2020-11-17 15:05:04 | 显示全部楼层 |阅读模式

马上注册,结交更多好友,享用更多功能,让你轻松玩转社区。

您需要 登录 才可以下载或查看,没有帐号?欢迎注册

x
本帖最后由 mathematica 于 2020-11-18 09:28 编辑

\[\frac{640320^{3/2}}{12\pi}=\frac{426880\sqrt{10005}}{\pi}=\sum^\infty_{k=0}\frac{(6k)!(545140134k+13591409)}{(3k)!(k!)^3\left(-640320\right)^{3k}}\]


https://en.wikipedia.org/wiki/Chudnovsky_algorithm
从维基百科偷来的
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2020-11-18 08:55:36 | 显示全部楼层
还有用gauss-勒让德    AGM-method计算圆周率的,但是估计消耗的内存非常巨大,以致于用上面的办法了
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2020-11-18 09:40:04 | 显示全部楼层
本帖最后由 mathematica 于 2020-11-18 09:48 编辑

拉马努金圆周率公式:
\[\frac{1}{\pi}=\frac{2\sqrt{2}}{99^2}\sum^\infty_{k=0}\frac{(4k)!(26390k+1103)}{(k!)^4396^{4k}}


\]


\[

\frac{1}{\pi}=\frac{2\sqrt{2}}{99^2}\sum^\infty_{k=0}\frac{(4k)!}{(k!)^4}\frac{(26390k+1103)}{396^{4k}}

\]
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2020-11-18 10:03:57 | 显示全部楼层
Super PI is a single threaded benchmark that calculates pi to a specific number of digits. It uses the Gauss-Legendre algorithm and is a Windows port of a program used by Yasumasa Kanada in 1995 to compute pi to 232 digits
http://www.superpi.net/About/
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2020-11-18 10:07:08 | 显示全部楼层
PiFast : the fastest windows program to compute pi
http://numbers.computation.free. ... Program/pifast.html
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2020-11-19 09:04:14 | 显示全部楼层
In Part 3 we managed to calculate 1,000,000 decimal places of π with Machin's arctan formula. Our stated aim was 100,000,000 places which we are going to achieve now!  Fun with Maths and Python  This is a having fun with maths and python article. See the introduction for important information!  We have still got a long way to go though, and we'll have to improve both our algorithm (formula) for π and our implementation.  The current darling of the π world is the Chudnovsky algorithm which is similar to the arctan formula but it converges much quicker. It is also rather complicated. The formula itself is derived from one by Ramanjuan who's work was extraordinary in the extreme. It isn't trivial to prove, so I won't! Here is Chudnovsky's formula for π as it is usually stated:
https://www.craig-wood.com/nick/articles/pi-chudnovsky/
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2020-11-19 09:11:49 | 显示全部楼层
https://arxiv.org/pdf/1809.00533.pdf
A detailed proof of the Chudnovsky formula with means of basic complex analysis
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 6 天前 | 显示全部楼层
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
您需要登录后才可以回帖 登录 | 欢迎注册

本版积分规则

小黑屋|手机版|数学研发网 ( 苏ICP备07505100号 )

GMT+8, 2020-12-3 21:49 , Processed in 0.063164 second(s), 16 queries .

Powered by Discuz! X3.4

© 2001-2017 Comsenz Inc.

快速回复 返回顶部 返回列表