The Computation of π to 29,360,000 Decimal Digits Using Borweins’ Quartically Convergent Algorithm
In a recent work , 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  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.
KeywordsFast Fourier Transform Discrete Fourier Transform Main Memory Chinese Remainder Theorem Decimal Digit
Unable to display preview. Download preview PDF.
- 3.A. Borodin & I. Munro, The Computational Complexity ofAlgebraic and Numeric Problems, American Elsevier, New York, 1975.Google Scholar
- 9..W. Gosper, private communication.Google Scholar
- 11.G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, London, 1984.Google Scholar
- 12.Y. Kanada & Y. Tamura, Calculation of π to 10, 013, 395 Decimal Places Based on the Gauss-Lependre Algorithm and Gauss Arctangent Relation, Computer Centre, University of Tokyo, 1983.Google Scholar
- 13.D. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, AddisonWesley, Reading, Mass., 1981.Google Scholar