计算圆周率的Chudnovsky公式

nyy

这个公式可不得了，之前拉马努金公式每计算一项可以获得8位有效数字已经很了不起了，然而他们的这个公式居然可以做到每计算一项得出15位有效数字！1994年，他们利用这个强大的公式，得到了圆周率小数点后40.44亿位。

https://baijiahao.baidu.com/s?id ... r=spider&for=pc

这儿声称8位、15位，很显然，我的计算结果打了他们的假！

nyy

拉马努金是怎么发现这个计算圆周率的公式的呢？

nyy

nyy 发表于 2023-12-21 11:21
https://baijiahao.baidu.com/s?id=1642938913706777216&wfr=spider&for=pc

这儿声称8位、15位，很显然 ...

https://math.stackexchange.com/q ... pproximation-for-pi

这儿给出了为什么每项是8个有效数字，实际上不到8个，但是非常接近8个的原因

nyy

nyy 发表于 2023-12-21 11:13
计算结果
-5.2991698741646948426044878234657120328437939087364571033028283320584\
567825035586620 ...

大约对了14196项，14196/1001=14.1818181818，平均每项增加14.18个有效数字。

这个14.18是怎么来的呢？
(6k+1)(6k+2)(6k+3)(6k+4)(6k+5)(6k+6)/((3k+1)(3k+2)(3k+3))/(k+1)^3/640320^3
这个值取极限，
得到a=2^3*6^3/640320^3,
然后对这个a的倒数取以10位底的对数，得到14.18164746位有效数字。

nyy

现在证明了 Chudnovsky 公式中的级数可以用 binary splitting 算法求得。我们整理一下公式

https://www.cnblogs.com/messier51/p/15872093.html


Pi - Chudnovsky
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/

nyy

这儿有不少马青公式

https://arxiv.org/pdf/2312.05413.pdf

nyy

nyy 发表于 2023-12-26 16:24
现在证明了 Chudnovsky 公式中的级数可以用 binary splitting 算法求得。我们整理一下公式

https://www.cn ...

binary splitting 算法，这个算法为什么能加速，有人知道为什么吗？