3,166 research outputs found
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions
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", to more general series. The
algorithm provides a rapid means of evaluating Li_s(z) for general values of
complex s and the region of complex z values given by |z^2/(z-1)|<4.
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.Comment: 37 pages, 6 graphs, 14 full-color phase plots. v3: Added discussion
of a fast Hurwitz algorithm; expanded development of the monodromy
v4:Correction and clarifiction of monodrom
Asymptotic and exact expansions of heat traces
We study heat traces associated with positive unbounded operators with
compact inverses. With the help of the inverse Mellin transform we derive
necessary conditions for the existence of a short time asymptotic expansion.
The conditions are formulated in terms of the meromorphic extension of the
associated spectral zeta-functions and proven to be verified for a large class
of operators. We also address the problem of convergence of the obtained
asymptotic expansions. General results are illustrated with a number of
explicit examples.Comment: 44 LaTeX pages, 2 figure
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
Numerical evaluation of multiple polylogarithms
Multiple polylogarithms appear in analytic calculations of higher order
corrections in quantum field theory. In this article we study the numerical
evaluation of multiple polylogarithms. We provide algorithms, which allow the
evaluation for arbitrary complex arguments and without any restriction on the
weight. We have implemented these algorithms with arbitrary precision
arithmetic in C++ within the GiNaC framework.Comment: 23 page
Convergence Acceleration Techniques
This work describes numerical methods that are useful in many areas: examples
include statistical modelling (bioinformatics, computational biology),
theoretical physics, and even pure mathematics. The methods are primarily
useful for the acceleration of slowly convergent and the summation of divergent
series that are ubiquitous in relevant applications. The computing time is
reduced in many cases by orders of magnitude.Comment: 6 pages, LaTeX; provides an easy-to-understand introduction to the
field of convergence acceleratio
On Differences of Zeta Values
Finite differences of values of the Riemann zeta function at the integers are
explored. Such quantities, which occur as coefficients in Newton series
representations, have surfaced in works of Maslanka, Coffey, Baez-Duarte, Voros
and others. We apply the theory of Norlund-Rice integrals in conjunction with
the saddle point method and derive precise asymptotic estimates. The method
extends to Dirichlet L-functions and our estimates appear to be partly related
to earlier investigations surrounding Li's criterion for the Riemann
hypothesis.Comment: 18 page
Simultaneous generation for zeta values by the Markov-WZ method
By application of the Markov-WZ method, we prove a more general form of a
bivariate generating function identity containing, as particular cases,
Koecher's and Almkvist-Granville's Ap\'ery-like formulae for odd zeta values.
As a consequence, we get a new identity producing Ap\'ery-like series for all
convergent at the geometric rate with ratio
Comment: 7 page
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