6,754 research outputs found
Time- and Space-Efficient Evaluation of Some Hypergeometric Constants
The currently best known algorithms for the numerical evaluation of
hypergeometric constants such as to decimal digits have time
complexity and space complexity of or .
Following work from Cheng, Gergel, Kim and Zima, we present a new algorithm
with the same asymptotic complexity, but more efficient in practice. Our
implementation of this algorithm improves slightly over existing programs for
the computation of , and we announce a new record of 2 billion digits for
Analytical expressions and numerical evaluation of the luminosity distance in a flat cosmology
Accurate and efficient methods to evaluate cosmological distances are an
important tool in modern precision cosmology. In a flat CDM cosmology,
the luminosity distance can be expressed in terms of elliptic integrals. We
derive an alternative and simple expression for the luminosity distance in a
flat CDM based on hypergeometric functions. Using a timing experiment
we compare the computation time for the numerical evaluation of the various
exact formulae, as well as for two approximate fitting formulae available in
the literature. We find that our novel expression is the most efficient exact
expression in the redshift range . Ideally, it can be combined with
the expression based on Carlson's elliptic integrals in the range
for high precision cosmology distance calculations over the entire redshift
range. On the other hand, for practical work where relative errors of about
0.1% are acceptable, the analytical approximation proposed by Adachi & Kasai
(2012) is a suitable alternative.Comment: 4 pages, 1 figure, accepted for publication in MNRA
Evaluating parametric holonomic sequences using rectangular splitting
We adapt the rectangular splitting technique of Paterson and Stockmeyer to
the problem of evaluating terms in holonomic sequences that depend on a
parameter. This approach allows computing the -th term in a recurrent
sequence of suitable type using "expensive" operations at the cost
of an increased number of "cheap" operations.
Rectangular splitting has little overhead and can perform better than either
naive evaluation or asymptotically faster algorithms for ranges of
encountered in applications. As an example, fast numerical evaluation of the
gamma function is investigated. Our work generalizes two previous algorithms of
Smith.Comment: 8 pages, 2 figure
Covariant Compton Scattering Kernel in General Relativistic Radiative Transfer
A covariant scattering kernel is a core component in any self-consistent
general relativistic radiative transfer formulation in scattering media. An
explicit closed-form expression for a covariant Compton scattering kernel with
a good dynamical energy range has unfortunately not been available thus far.
Such an expression is essential to obtain numerical solutions to the general
relativistic radiative transfer equations in complicated astrophysical settings
where strong scattering effects are coupled with highly relativistic flows and
steep gravitational gradients. Moreover, this must be performed in an efficient
manner. With a self-consistent covariant approach, we have derived a
closed-form expression for the Compton scattering kernel for arbitrary energy
range. The scattering kernel and its angular moments are expressed in terms of
hypergeometric functions, and their derivations are shown explicitly in this
paper. We also evaluate the kernel and its moments numerically, assessing
various techniques for their calculation. Finally, we demonstrate that our
closed-form expression produces the same results as previous calculations,
which employ fully numerical computation methods and are applicable only in
more restrictive settings.Comment: 29 pages, 10 figures, 2 tables; Accepted for publication in MNRA
Recursive Estimation of Orientation Based on the Bingham Distribution
Directional estimation is a common problem in many tracking applications.
Traditional filters such as the Kalman filter perform poorly because they fail
to take the periodic nature of the problem into account. We present a recursive
filter for directional data based on the Bingham distribution in two
dimensions. The proposed filter can be applied to circular filtering problems
with 180 degree symmetry, i.e., rotations by 180 degrees cannot be
distinguished. It is easily implemented using standard numerical techniques and
suitable for real-time applications. The presented approach is extensible to
quaternions, which allow tracking arbitrary three-dimensional orientations. We
evaluate our filter in a challenging scenario and compare it to a traditional
Kalman filtering approach
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