梅钦类公式

梅钦类公式（英语：Machin-like formula）是数学中计算圆周率的一个常用技巧，它是梅钦公式的推广，梅钦公式的形式为

{\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}

梅钦依据此公式，把圆周率计算到一百多位小数。

梅钦类公式的形式为：

$c_{0}{\frac {\pi }{4}}=\sum _{n=1}^{N}c_{n}\arctan {\frac {a_{n}}{b_{n}}}$

(1)

其中， $a_{n}$ 和 $b_{n}$ 为正整数，且 $a_{n}<b_{n}$ ， $c_{n}$ 为非零整数，且 $c_{0}$ 为正整数。

梅钦类公式的应用可结合反正切函数的泰勒级数展开：

$\arctan x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+...$

(4)

导出

根据角的和差公式，

\sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta

\cos(\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta

若 $-{\frac {\pi }{2}}<\arctan {\frac {a_{1}}{b_{1}}}+\arctan {\frac {a_{2}}{b_{2}}}<{\frac {\pi }{2}}.$ 有

$\arctan {\frac {a_{1}}{b_{1}}}+\arctan {\frac {a_{2}}{b_{2}}}=\arctan {\frac {a_{1}b_{2}+a_{2}b_{1}}{b_{1}b_{2}-a_{1}a_{2}}},$

(2)

反复应用这一方程，可得到所有的梅钦类公式，比如最初的梅钦公式：

2\arctan {\frac {1}{5}}

=\arctan {\frac {1}{5}}+\arctan {\frac {1}{5}}

=\arctan {\frac {1\times 5+1\times 5}{5\times 5-1\times 1}}

=\arctan {\frac {10}{24}}

=\arctan {\frac {5}{12}}

4\arctan {\frac {1}{5}}

=2\arctan {\frac {1}{5}}+2\arctan {\frac {1}{5}}

=\arctan {\frac {5}{12}}+\arctan {\frac {5}{12}}

=\arctan {\frac {5\times 12+5\times 12}{12\times 12-5\times 5}}

=\arctan {\frac {120}{119}}

4\arctan {\frac {1}{5}}-{\frac {\pi }{4}}

=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{1}}

=4\arctan {\frac {1}{5}}+\arctan {\frac {-1}{1}}

=\arctan {\frac {120}{119}}+\arctan {\frac {-1}{1}}

=\arctan {\frac {120\times 1+(-1)\times 119}{119\times 1-120\times (-1)}}

=\arctan {\frac {1}{239}}

{\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}

用梅钦公式编程计算圆周率(C++)

#include<stdio.h>
#include<iostream>
using namespace std;
int main(void)
{   //本程序每四位数输出，如果请求计算的位数不是4的整数倍，最后输出可能会少1~3位数
	long a[2]={956,80},b[2]={57121,25},i=0,j,k,p,q,r,s=2,t,u,v,N,M=10000;
	printf("%9cMachin%6cpi=16arctan(1/5)-4arctan(1/239)\nPlease input a number.\n",32,32);
	cin>>N,N=N/4+3;
	long *pi=new long[N],*e=new long[N];
	while(i<N)pi[i++]=0;
	while(--s+1)
	{
		for(*e=a[k=s],i=N;--i;)e[i]=0;
		for(q=1;j=i-1,i<N;e[i]?0:++i,q+=2,k=!k)
			for(r=v=0;++j<N;pi[j]+=k?u:-u)u=(t=v*M+(e[j]=(p=r*M+e[j])/b[s]))/q,r=p%b[s],v=t%q;
	}
	while(--i)(pi[i]=(t=pi[i]+s)%M)<0?pi[i]+=M,s=t/M-1:s=t/M;
	for(cout<<"3.";++i<N-2;)printf("%04ld",pi[i]);
	delete []pi,delete []e,cin.ignore(),cin.ignore();
	return 0;
}

参考文献

埃里克·韦斯坦因. Machin-like formulas. MathWorld.
The constant π （页面存档备份，存于互联网档案馆）
Machin's Merit （页面存档备份，存于互联网档案馆） at MathPages
Archimedes' constant pi - Machin's formula gives a proof for the John Machin`s formula
GUIDORIZZI, Hamilton Luiz. Um Curso de Cálculo, vol. 4. 3ª edição. Rio de Janeiro: Livros Técnicos e Científicos Editora, 1999.