Numerical Algorithms

, Volume 47, Issue 3, pp 211–252

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

Original Paper

DOI: 10.1007/s11075-007-9153-8

Cite this article as:
Vepštas, L. Numer Algor (2008) 47: 211. doi:10.1007/s11075-007-9153-8

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 

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.AustinUSA

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