124,681 research outputs found
Scaling of the Critical Function for the Standard Map: Some Numerical Results
The behavior of the critical function for the breakdown of the homotopically
non-trivial invariant (KAM) curves for the standard map, as the rotation number
tends to a rational number, is investigated using a version of Greene's residue
criterion. The results are compared to the analogous ones for the radius of
convergence of the Lindstedt series, in which case rigorous theorems have been
proved. The conjectured interpolation of the critical function in terms of the
Bryuno function is discussed.Comment: 26 pages, 3 figures, 13 table
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
Fast computation of Bernoulli, Tangent and Secant numbers
We consider the computation of Bernoulli, Tangent (zag), and Secant (zig or
Euler) numbers. In particular, we give asymptotically fast algorithms for
computing the first n such numbers in O(n^2.(log n)^(2+o(1))) bit-operations.
We also give very short in-place algorithms for computing the first n Tangent
or Secant numbers in O(n^2) integer operations. These algorithms are extremely
simple, and fast for moderate values of n. They are faster and use less space
than the algorithms of Atkinson (for Tangent and Secant numbers) and Akiyama
and Tanigawa (for Bernoulli numbers).Comment: 16 pages. To appear in Computational and Analytical Mathematics
(associated with the May 2011 workshop in honour of Jonathan Borwein's 60th
birthday). For further information, see
http://maths.anu.edu.au/~brent/pub/pub242.htm
Constrained correlation functions from the Millennium Simulation
Context. In previous work, we developed a quasi-Gaussian approximation for
the likelihood of correlation functions, which, in contrast to the usual
Gaussian approach, incorporates fundamental mathematical constraints on
correlation functions. The analytical computation of these constraints is only
feasible in the case of correlation functions of one-dimensional random fields.
Aims. In this work, we aim to obtain corresponding constraints in the case of
higher-dimensional random fields and test them in a more realistic context.
Methods. We develop numerical methods to compute the constraints on
correlation functions which are also applicable for two- and three-dimensional
fields. In order to test the accuracy of the numerically obtained constraints,
we compare them to the analytical results for the one-dimensional case.
Finally, we compute correlation functions from the halo catalog of the
Millennium Simulation, check whether they obey the constraints, and examine the
performance of the transformation used in the construction of the
quasi-Gaussian likelihood.
Results. We find that our numerical methods of computing the constraints are
robust and that the correlation functions measured from the Millennium
Simulation obey them. Despite the fact that the measured correlation functions
lie well inside the allowed region of parameter space, i.e. far away from the
boundaries of the allowed volume defined by the constraints, we find strong
indications that the quasi-Gaussian likelihood yields a substantially more
accurate description than the Gaussian one.Comment: 11 pages, 13 figures, updated to match version accepted by A&
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