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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. H. Bailey, “A high-performance fast Fourier transform algorithm for the Cray-2,” J. Supercomputing, v. 1, 1987, pp. 43–60.
P. Beckmann, A History of Pi, Golem Press, Boulder, Co, 1971.
A. Borodin & I. Munro, The Computational Complexity ofAlgebraic and Numeric Problems, American Elsevier, New York, 1975.
J. M. Borwein & P. B. Borwein, “The arithmetic-geometric mean and fast computation of elementary functions,“ Siam Rev., v. 26, 1984, pp. 351–366.
J. M. Borwein & P. B. Borwein, “More quadratically converging algorithms for π,” Math. Comp., v. 46, 1986, pp. 247–253.
J. M. Borwein & P. B. Borwein, Pi and the Agm—A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987.
>R. P. Brent, “Fast multiple-precision evaluation of elementary functions,” J. Assoc. Comput. Mach., v. 23, 1976, pp. 242–251.
E. O. Brigham, The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, N. J., 1974.
W. Gosper, private communication.
Emil Grosswald, Topics from the Theory of Numbers, Macmillan, New York, 1966.
G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, London, 1984.
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.
D. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, AddisonWesley, Reading, Mass., 1981.
E. Salamin, “Computation of π using arithmetic-geometric mean,” Math. Comp., v. 30, 1976, pp. 565–570.
D. Shanks & J. W. Wrench Jr., “Calculation of π to 100,000 decimals,” Math. Comp., v. 16, 1962, pp. 76–99.
P. Swarztrauber, “Fft algorithms for vector computers,” Parallel Comput., v. 1, 1984, pp. 45–64.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Science+Business Media New York
About this chapter
Cite this chapter
Bailey, D.H. (2004). The Computation of π to 29,360,000 Decimal Digits Using Borweins’ Quartically Convergent Algorithm. In: Pi: A Source Book. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4217-6_60
Download citation
DOI: https://doi.org/10.1007/978-1-4757-4217-6_60
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-1915-1
Online ISBN: 978-1-4757-4217-6
eBook Packages: Springer Book Archive