The computation of $\pi$ to $29,360,000$ decimal digits using Borweins’ quartically convergent algorithm

Abstract:

In a recent work [6], Borwein and Borwein derived a class of algorithms based on the theory of elliptic integrals that yield very rapidly convergent approximations to elementary constants. The author has implemented Borweins’ quartically convergent algorithm for $1/\pi$, using a prime modulus transform multi-precision technique, to compute over 29,360,000 digits of the decimal expansion of $\pi$. The result was checked by using a different algorithm, also due to the Borweins, that converges quadratically to $\pi$. These computations were performed as a system test of the Cray-2 operated by the Numerical Aerodynamical Simulation (NAS) Program at NASA Ames Research Center. The calculations were made possible by the very large memory of the Cray-2. Until recently, the largest computation of the decimal expansion of $\pi$ was due to Kanada and Tamura [12] of the University of Tokyo. In 1983 they computed approximately 16 million digits on a Hitachi S-810 computer. Late in 1985 Gosper [9] reported computing 17 million digits using a Symbolics workstation. Since the computation described in this paper was performed, Kanada has reported extending the computation of $\pi$ to over 134 million digits (January 1987). This paper describes the algorithms and techniques used in the author’s computation, both for converging to $\pi$ and for performing the required multi-precision arithmetic. The results of statistical analyses of the computed decimal expansion are also included.