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The Computation of pi to 29,360,000 Decimal Digits Using Borweins' Quartically Convergent Algorithm

By David H. Bailey


In a recent work [6], Borwein and Borwein derived a class of algorithms based on the theory of complete elliptic integrals that yield very rapidly convergent approximations to elementary constants. The author has implemented Borweins ' quartically convergent al-gorithm for l/r, using a prime modulus transform multi-precision technique, to compute over 29,360,000 digits of the decimal expansion of r. The result was checked by using a different algorithm, also due to the Borweins, that converges quadratically to r. These computations were performed as a system test of the Cray-2 operated by the Numerical Aerodynamic Simulation (NAS) Program at NASA Ames Research Center. The calcula-tions were made possible by the very large main memory of the Cray-2. Until recently the largest computation of the decimal expansion of r 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 r to over 134 million digits (January 1987). This paper describes the algorithms and techniques used in the author's computation, both for converging to 7r and for performing the required multi-precision arithmetic. The results of statistical analyses of the computed decimal expansion are also included

Year: 1987
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