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

Content Provider	Springer Nature Link
Author	Vepštas, Linas
Copyright Year	2008
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 Li$_{ s }$(z) for general values of complex s and a kidney-shaped region of complex z values given by ∣z $^{2}$/(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.
Ending Page	252
Page Count	42
Starting Page	211
File Format	PDF
ISSN	10171398
e-ISSN	15729265
Journal	Numerical Algorithms
Issue Number	3
Volume Number	47
Language	English
Publisher	Springer US
Publisher Date	2008-03-04
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Series acceleration Hurwitz and Lerch zeta functions Numeric Computing Theory of Computation Mathematics Algorithm Hurwitz zeta function Monodromy Polylogarithm Numerical approximation and evaluation Summation of series Algebra Algorithms Analytic computations Approximation algorithms
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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