Pi: A Source Book pp 562-575 | Cite as

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

## 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.

## Keywords

Fast Fourier Transform Discrete Fourier Transform Main Memory Chinese Remainder Theorem Decimal Digit## Preview

Unable to display preview. Download preview PDF.

## References

- 1.D. H. Bailey, “A high-performance fast Fourier transform algorithm for the Cray-2,” J. Supercomputing, v. 1, 1987, pp. 43–60.zbMATHCrossRefGoogle Scholar
- 2.
- 3.A. Borodin & I. Munro,
*The Computational Complexity of Algebraic and Numeric Problems*, American Elsevier, New York, 1975.zbMATHGoogle Scholar - 4.J. M. Borwein & P. B. Borwein, “The arithmetic-geometric mean and fast computation of elementary functions,”
*SIAM Rev*.,*v*. 26, 1984, pp. 351–366.MathSciNetzbMATHCrossRefGoogle Scholar - 5.J. M. Borwein & P. B. Borwein, “More quadratically converging algorithms for w,” Math.
*Comp*.,*v*. 46, 1986, pp. 247–253.MathSciNetzbMATHGoogle Scholar - 6.J. M. Borwein & P. B. Borwein,
*Pi and the AGM-A Study in Analytic Number Theory and Computational Complexity*, Wiley, New York, 1987.zbMATHGoogle Scholar - 7.R. P. Brent, “Fast multiple-precision evaluation of elementary functions,”
*J. Assoc. Comput. Mach*., v. 23, 1976, pp. 242–251.MathSciNetzbMATHCrossRefGoogle Scholar - 8.E. O. Brigham,
*The Fast Fourier Transform*, Prentice-Hall, Englewood Cliffs, N. J., 1974.Google Scholar - 9.W. Gosper, private communication.Google Scholar
- 10.
- 11.G. H. Hardy dc 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 w to*10,013,395*Decimal Places Based on the Gauss-Legendre 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, Addison-Wesley, Reading, Mass., 1981.zbMATHGoogle Scholar - 14.E. Salamin, “Computation of w using arithmetic-geometric
*mean*,*” Math*. Comp., v. 30, 1976, pp. 565–570.MathSciNetzbMATHGoogle Scholar - 15.D. Shanks & J. W. Wrench, JR., “Calculation of w to 100,000 decimals,”
*Math. Comp*., v. 16, 1962, pp. 76–99.MathSciNetzbMATHGoogle Scholar - 16.P. Swarztrauber, “FFT algorithms for vector computers,” Parallel
*Comput*.,*v*. 1, 1984, pp. 45–64.zbMATHCrossRefGoogle Scholar