die von Jonathan Michael Borwein und Peter Benjamin Borwein entwickelten Algorithmen zur schnellen Berechnung von Näherungen zu π.

1984 fanden die Borweins mit Hilfe des Gaußschen arithmetisch-geometrischen Mittels die Iteration

\begin{eqnarray}{\alpha }_{0}=\sqrt{2},\text{\hspace{0.17em}}{\beta }_{0}=0,\text{\hspace{0.17em}},{\pi }_{0}=2+\sqrt{2}\\ {\alpha }_{n+1}=\frac{1}{2}(\sqrt{{\alpha }_{n}}+\frac{1}{\sqrt{{\alpha }_{n}}}),\\ {\beta }_{n+1}=\sqrt{{\alpha }_{n}}\frac{{\beta }_{n}+1}{{\beta }_{n}+{\alpha }_{n}},{\pi }_{n+1}={\pi }_{n}{\beta }_{n+1}\frac{1+{\alpha }_{n+1}}{1+{\beta }_{n+1}},\end{eqnarray}

\begin{eqnarray}{y}_{0}=\frac{1}{\sqrt{2}},\text{\hspace{1em}}\text{\hspace{1em}}{\alpha }_{0}=\frac{1}{2},\\ {y}_{n+1}=\frac{1-\sqrt{1-{y}_{n}^{2}}}{1+\sqrt{1-{y}_{n}^{2}}},\\ {\alpha }_{n+1}={(1+{y}_{n+1})}^{2}{\alpha }_{n}-2\cdot 2{y}_{n+1},\end{eqnarray}

\begin{eqnarray}{y}_{0}=\frac{\sqrt{3}-1}{2},\text{\hspace{1em}}\text{\hspace{1em}}{\alpha }_{0}=\frac{1}{3},\\ {y}_{n+1}=\frac{1-\sqrt[3]{1-{y}_{n}^{3}}}{1+2\sqrt[3]{1-{y}_{n}^{3}}},\\ {\alpha }_{n+1}={(1+2{y}_{n+1})}^{2}{\alpha }_{n}-4\cdot {3}^{n}{y}_{n+1}(1+{y}_{n+1}),\end{eqnarray}

\begin{eqnarray}{y}_{0}=\sqrt{2}-1,\text{\hspace{1em}}\text{\hspace{1em}}{\alpha }_{0}=6-4\sqrt{2}\\ {y}_{n+1}=\frac{1-\sqrt[4]{1-{y}_{n}^{4}}}{1+\sqrt[4]{1-{y}_{n}^{4}}},\\ {\alpha }_{n+1}={(1+{y}_{n+1})}^{4}{\alpha }_{n}-8\cdot {4}^{n}(1+{y}_{n+1}+{y}_{n+1}^{2}),\end{eqnarray}

\begin{eqnarray}\frac{1}{{\alpha }_{0}}=2.91421356237309504880168872420\ldots \\ \frac{1}{{\alpha }_{1}}=3.14159264621354228214934443198\ldots \\ \frac{1}{{\alpha }_{2}}=3.14159265358979323846264338327\ldots \end{eqnarray}