梅欽類公式

梅欽類公式（英語：Machin-like formula）是數學中計算圓周率的一個常用技巧，它是梅欽公式的推廣，梅欽公式的形式為

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

梅欽依據此公式，把圓周率計算到一百多位小數。

梅欽類公式的形式為：

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

1

其中， ${\displaystyle a_{n}}$ 和 ${\displaystyle b_{n}}$ 為正 整數，且 ${\displaystyle a_{n}<b_{n}}$， ${\displaystyle c_{n}}$ 為非零整數，且${\displaystyle c_{0}}$ 為正整數。

梅欽類公式的應用可結合反正切函數的泰勒級數展開：

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

4

導出

根據角的和差公式，

${\displaystyle \sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta }$

${\displaystyle \cos(\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta }$

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

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

2

反覆應用這一方程，可得到所有的梅欽類公式，比如最初的梅欽公式：

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

用梅欽公式編程計算圓周率(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;
}

參考文獻

• 埃里克·韋斯坦因. Machin-like formulas. MathWorld.
• The constant π （頁面存檔備份，存於網際網路檔案館）
• Machin's Merit （頁面存檔備份，存於網際網路檔案館） 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.