A fast algorithm based on the arithmetic-geometric mean (Q1974750)

\textit{J. M. Borwein} and \textit{P. B. Borwein}'s algorithm [SIAM Rev. 26, 351-366 (1984; Zbl 0557.65009); \(\pi\) and the AGM. A study in analytic number theory and computational complexity (1987; Zbl 0611.10001)] for calculation of \(\pi\) is analyzed in detail. The algorithm is based on differentiation with respect to a parameter in the recursive relations that define Gauss arithmetic-geometric mean sequences. A more natural substantiation of the convergence of the algorithm (without any deviation into the complex plane) is presented. The authors give a more accurate estimate for the rate of convergence: \[ \pi_n- \pi< 10^{-2^{n+1}- 2^{n-1}+2}. \]