An algorithm for computing $\pi^{-1}$ Wikipedia: Borwein's algorithm claims > Start out by setting > > $$\begin{align} a_0 & = \frac{1}{3} \\\ s_0 & = \frac{\sqrt{3} - 1}{2} \end{align} $$ > > Then iterate > > $$ \begin{align} r_{k+1} & = \frac{3}{1 + 2(1-s_k^3)^{1/3}} \\\ s_{k+1} & = \frac{r_{k+1} - 1}{2} \\\ a_{k+1} & = r_{k+1}^2 a_k - 3^k(r_{k+1}^2-1) \end{align} $$ > > **Then $a_k$ converges cubically to $1/\pi$;** that is, each iteration approximately triples the number of correct digits. sadly without direct reference or proof. Where (precise reference) can I find the proof, or even better, could it be summarized in an answer? (I'm more interested in seeing _that_ the recurrence relation converges to $1/\pi$ than in the rate of convergence.)

Is this equivalent to Theorem 6.1 here? The reference is to J.M. Borwein and P.B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), 691--701.