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The Computation of π to 29,360,000 Decimal Digits Using Borweins’ Quartically Convergent Algorithm

  • David H. Bailey
Chapter

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/π, using a prime modulus transform multi-precision technique, to compute over 29,360,000 digits of the decimal expansion of π. The result was checked by using a different algorithm, also due to the Borweins, that converges quadratically to π. 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 π 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 π to over 134 million digits (January 1987).

This paper describes the algorithms and techniques used in the author’s computation, both for converging to π and for performing the required multi-precision arithmetic. The results of statistical analyses of the computed decimal expansion are also included.

Keywords

Fast Fourier Transform Discrete Fourier Transform Main Memory Chinese Remainder Theorem Decimal Digit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • David H. Bailey
    • 1
  1. 1.NAS Systems DivisionNASA Ames Research CenterMoffett FieldUSA

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