On the rapid computation of various polylogarithmic constants
HTML articles powered by AMS MathViewer
- by David Bailey, Peter Borwein and Simon Plouffe PDF
- Math. Comp. 66 (1997), 903-913 Request permission
Abstract:
We give algorithms for the computation of the $d$-th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of $\log {(2)}$ or $\pi$ on a modest work station in a few hours run time. We demonstrate this technique by computing the ten billionth hexadecimal digit of $\pi$, the billionth hexadecimal digits of $\pi ^{2}, \; \log (2)$ and $\log ^{2}(2)$, and the ten billionth decimal digit of $\log (9/10)$. These calculations rest on the observation that very special types of identities exist for certain numbers like $\pi$, $\pi ^{2}$, $\log (2)$ and $\log ^{2}(2)$. These are essentially polylogarithmic ladders in an integer base. A number of these identities that we derive in this work appear to be new, for example the critical identity for $\pi$: \begin{equation*}\pi = \sum _{i=0}^{\infty }\frac {1}{16^{i}}\bigr ( \frac {4}{8i+1} - \frac {2}{8i+4} - \frac {1}{8i+5} - \frac {1}{8i+6} \bigl ).\end{equation*}References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1966. MR 0208797
- V. Adamchik and S. Wagon, Pi: A 2000-year search changes direction (preprint).
- Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman, The design and analysis of computer algorithms, Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Second printing. MR 0413592
- David H. Bailey, Jonathan M. Borwein, and Roland Girgensohn, Experimental evaluation of Euler sums, Experiment. Math. 3 (1994), no. 1, 17–30. MR 1302815, DOI 10.1080/10586458.1994.10504573
- Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1987. A study in analytic number theory and computational complexity; A Wiley-Interscience Publication. MR 877728
- J. M. Borwein and P. B. Borwein, On the complexity of familiar functions and numbers, SIAM Rev. 30 (1988), no. 4, 589–601. MR 967961, DOI 10.1137/1030134
- J. M. Borwein, P. B. Borwein, and D. H. Bailey, Ramanujan, modular equations, and approximations to pi, or How to compute one billion digits of pi, Amer. Math. Monthly 96 (1989), no. 3, 201–219. MR 991866, DOI 10.2307/2325206
- Richard P. Brent, The parallel evaluation of general arithmetic expressions, J. Assoc. Comput. Mach. 21 (1974), 201–206. MR 660280, DOI 10.1145/321812.321815
- Stephen A. Cook, A taxonomy of problems with fast parallel algorithms, Inform. and Control 64 (1985), no. 1-3, 2–22. MR 837088, DOI 10.1016/S0019-9958(85)80041-3
- R. Crandall, K. Dilcher, and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp. 66 (1997), 433–449.
- Richard E. Crandall and Joe P. Buhler, On the evaluation of Euler sums, Experiment. Math. 3 (1994), no. 4, 275–285. MR 1341720, DOI 10.1080/10586458.1994.10504297
- H. R. P. Ferguson and D. H. Bailey, Analysis of PSLQ, an integer relation algorithm (preprint).
- E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975.
- Donald E. Knuth, The art of computer programming. Vol. 2, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms. MR 633878
- Leonard Lewin, Polylogarithms and associated functions, North-Holland Publishing Co., New York-Amsterdam, 1981. With a foreword by A. J. Van der Poorten. MR 618278
- Leonard Lewin (ed.), Structural properties of polylogarithms, Mathematical Surveys and Monographs, vol. 37, American Mathematical Society, Providence, RI, 1991. MR 1148371, DOI 10.1090/surv/037
- N. Nielsen, Der Eulersche Dilogarithmus, Halle, Leipzig, 1909.
- Stanley Rabinowitz and Stan Wagon, A spigot algorithm for the digits of $\pi$, Amer. Math. Monthly 102 (1995), no. 3, 195–203. MR 1317842, DOI 10.2307/2975006
- Arnold Schönhage, Asymptotically fast algorithms for the numerical multiplication and division of polynomials with complex coefficients, Computer algebra (Marseille, 1982) Lecture Notes in Comput. Sci., vol. 144, Springer, Berlin-New York, 1982, pp. 3–15. MR 680048
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
- Herbert S. Wilf, Algorithms and complexity, Prentice Hall, Inc., Englewood Cliffs, NJ, 1986. MR 897317
Additional Information
- David Bailey
- Affiliation: NASA Ames Research Center, Mail Stop T27A-1, Moffett Field, California 94035-1000
- MR Author ID: 29355
- Email: dbailey@nas.nasa.gov
- Peter Borwein
- Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6
- Email: pborwein@cecm.sfu.ca
- Simon Plouffe
- Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6
- Email: plouffe@cecm.sfu.ca
- Received by editor(s): October 11, 1995
- Received by editor(s) in revised form: February 16, 1996
- Additional Notes: Research of the second author was supported in part by NSERC of Canada.
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp. 66 (1997), 903-913
- MSC (1991): Primary 11A05, 11Y16, 68Q25
- DOI: https://doi.org/10.1090/S0025-5718-97-00856-9
- MathSciNet review: 1415794