Numerical Algorithms

, Volume 47, Issue 3, pp 211–252 | Cite as

An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

Original Paper

Abstract

This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein’s “An efficient algorithm for computing the Riemann zeta function” by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating Lis(z) for general values of complex s and a kidney-shaped region of complex z values given by ∣z2/(z–1)∣<4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler–Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler–Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor’s series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.

Keywords

Polylogarithm Hurwitz zeta function Algorithm Monodromy Series acceleration 

Mathematics Subject Classifications (2000)

65B10 (primary) 11M35 11Y35 33F05 68W25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions, 10th Printing ed. Dover Publications (1972)Google Scholar
  2. 2.
    Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)MATHGoogle Scholar
  3. 3.
    Balanzario, E.P.: A generalized Euler–Maclaurin formula for the Hurwitz zeta function. Math. Slovaca 56(3), 307–316 (2006)MATHMathSciNetGoogle Scholar
  4. 4.
    Bloch, S.: Structural properties of polylogarithms. Mathematical surveys and monographs, vol. 37, ch. “Introduction to Higher Logarithms”, pp. 275–285. American Mathematical Society (1991)Google Scholar
  5. 5.
    Boas, R.P., Creighton Buck, R.: Polynomial Expansions of Analytic Functions. Academic Press Inc (1964)Google Scholar
  6. 6.
    Borwein, P.: An efficient algorithm for the Riemann zeta function. Constructive experimental and nonlinear analysis, CMS Conference Proceedings 27, pp. 29–34 (preprint, January 1995) (2000) http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
  7. 7.
    Brezinski, C.: Rational approximation to formal power series. J. Approx. Theory 25, 295–317 (1979)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Brezinski, C., Zaglia, M.R.: Extrapolation Methods. North Holland (1991)Google Scholar
  9. 9.
    Costin, O., Garoufalidis, S.: Resurgence of the fractional polylogarithms (2007) arXiv:math. CA/0701743Google Scholar
  10. 10.
    Crandall, R.E.: Note on fast polylogarithm computation. http://people.reed.edu/~crandall/papers/Polylog.pdf (January 2006)
  11. 11.
    Martin, D.W., Miller, G.F., Olver, F.J.W., Clenshaw, C.W., Goodwin, E.T., Wilkinson, J.W. (eds.): Modern Computing Methods, 2nd edn. Philosophical Library (1961)Google Scholar
  12. 12.
    Kirtman, B., Weniger, E.J.: Extrapolation methods for improving the convergence of oligomer calculations to the infinite chain limit of quasi-onedimensional stereoregular polymers. Comput. Math. Appl. 45, 189–215 (2003) arXiv:math.NA/0004115Google Scholar
  13. 13.
    Free Software Foundation: Gnu multiple precision arithmetic library. http://www.swox.com/gmp/
  14. 14.
    Graves-Morris, Jr. P.R., Baker, G.A.: Padé Approximants. Cambridge University Press (1996)Google Scholar
  15. 15.
    Hain, R.M.: Classical polylogarithms (1992) arXiv:alg-geom/9202022Google Scholar
  16. 16.
    Hasse, H.: Ein Summierungsverfahren fur die Riemannsche ζ-Reihe. Math. Z. 32, 458–464 (1930)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Crandall, R.E., Borwein, J.M., Bradley, D.M.: Computational strategies for the Riemann zeta function. J. Comput. Appl. Math. 121, 247–296 (2000) http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf Google Scholar
  18. 18.
    Jonquière, A.: Notes sur la série (polylogarithm). Bull. Soc. Math. France 17, 142–152 (1889)MathSciNetGoogle Scholar
  19. 19.
    Karatsuba, E.A.: Fast evaluation of the Hurwitz zeta function and Dirichlet L-series. Problem. Peredachi Informat. 34(4), 342–353 (1998)MATHMathSciNetGoogle Scholar
  20. 20.
    Lewin, L.: Polylogarithms and Associated Functions. North Holland (1981)Google Scholar
  21. 21.
    Hain, R.M., Macpherson, R.: Structural properties of polylogarithms. Mathematical Surveys and Monographs, vol. 37, ch. “Introduction to Higher Logarithms”, pp. 337–353. American Mathematical Society (1991)Google Scholar
  22. 22.
    Milne-Thomson, A.M.: The Calculus of Finite Differences. Chelsea Publishing (1933)Google Scholar
  23. 23.
    Bateman Manuscript Project: Higher transcendental functions, vol. 1. McGraw Hill, New York (1953)Google Scholar
  24. 24.
    Šleževičiene, R.: An efficient algorithm for computing Dirichlet L-functions. Integral Transforms Spec. Funct. 15(6), 513–522 (2004)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Soff, G., Joachim Weniger, E., Jentschura, U.D., Mohr, P.J.: Convergence acceleration via combined nonlinear-condensation transformations. Comput. Phys. Commun. 116, 28–54 (1999) arXiv:math.NA/9809111MATHCrossRefGoogle Scholar
  26. 26.
    Cohen, H., Villegas, F.R., Zagier, D.: Convergence acceleration of alternating series. Exp. Math. 9(1), 3–12 (2000) http://www.math.utexas.edu/~villegas/publications/conv-accel.pdf MATHGoogle Scholar
  27. 27.
    Weniger, E.J.: Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep. 10, 189–371 (1989) arXiv math.NA/0306302CrossRefGoogle Scholar
  28. 28.
    Weniger, E.J.: Algorithms for approximation, ch. Asymptotic approximations to truncation errors of series representations for special functions, pp. 331–348, Springer-Verlag (2007) arXiv:math/0511074Google Scholar
  29. 29.
    Vetterling, W.T., Press, W.H., Teukolsky, S.A., Flannery, B.P.: Numerical Recipes in C, the Art of Scientific Computing, 2nd edn. Cambidge University Press (1988)Google Scholar
  30. 30.
    Wood, D.: The computation of polylogarithms. Tech. Report 15-92*, University of Kent, Computing Laboratory, University of Kent, Canterbury, UK (June 1992) http://www.cs.kent.ac.uk/pubs/1992/110

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.AustinUSA

Personalised recommendations