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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

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


Author: David H. Bailey
Journal: Math. Comp. 50 (1988), 283-296
MSC: Primary 11Y60; Secondary 11-04, 11K16, 65-04
MathSciNet review: 917836
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1988-0917836-3
PII: S 0025-5718(1988)0917836-3
Article copyright: © Copyright 1988 American Mathematical Society