# Pi (AGM method) Calculator

## Calculates circular constant Pi using the Arithmetic-geometric mean method (AGM).

 $\pi={\large\frac{2\hspace{1}{\rm AGM}^2\left(1,{\large\frac{1}{\sqrt{2}}}\right)}{1-{\large\sum_{\small k=0}^{\small \infty}}2^kc_k^2}}\\\vspace{20}\\$
method
 2nd convergence (Salamin–Brent) 4th convergence (Borwein) 9th convergence
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 The calculation ends when two consecutive results are the same.The accuracy of π improves by increasing the number of digits for calculation.In 1976, Salamin and Brent discovered the new algorithm for calculating Pi based on the Gauss’s AGM formula (1809). The algorithm is quadratically convergent and each step of the algorithm doubles the number of correct digits. $\normal Gauss-Legendre\ method\ 1809\\\hspace{80}\pi={\large\frac{2\hspace{1}{\rm AGM}^2\left(1,{\large\frac{1}{\sqrt{2}}}\right)}{1-{\large\sum_{\small k=0}^{\small \infty}}2^kc_k^2}}\\\vspace{20}\\The\ Square\ {\rm AGM}\\\hspace{80} by\ \ Salamin\ &\ Brent,\ 1976\\(1)\ a_0=1,\hspace{20}b_0={\large\frac{1}{\sqrt{2}}},\hspace{20}t_0={\large\frac{1}{4}}\\(2)\ a_{n+1}={\large\frac{1}{2}}(a_n+b_n),\hspace{20}b_{n+1}=\sqrt{a_nb_n},\\\hspace{50}t_{n+1}=t_n-2^n(a_n-a_{n+1})^2\\(3)\ \pi=\lim_{n\to\infty}{\large\frac{(a_n+b_n)^2}{4t_n}}\\\vspace{20}\normal The\ Quartic\ {\rm AGM}\ by\ J.M.\ Borwein\\\hspace{120} &\ P.B.\ Borwein,\ 1985\\(1)\ y_0=\sqrt{2}-1,\hspace{20}a_0=6-4\sqrt{2}\\(2)\ y_{n}={\large\frac{1-\sqrt[4]{1-y_{n-1}^4}}{1+\sqrt[4]{1-y_{n-1}^4}}},\\\hspace{10}a_{n}=(1+y_{n})^4a_{n-1}-2^{2n+1}y_{n}(1+y_{n}+y_{n}^2)\\(3)\ \pi={\large\lim_{\small n \to\infty}\frac{1}{a_n}}\\$

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