J/A+A/619/A129 HKE-SDG code to solve hyperbolic Kepler eq. (Raposo-Pulido+ 2018)
An efficient code to solve the Kepler equation Hyperbolic case.
Raposo-Pulido V., Pelaez J.
<Astron. Astrophys. 619, A129 (2018)>
=2018A&A...619A.129R 2018A&A...619A.129R (SIMBAD/NED BibCode)
ADC_Keywords: Models ; Positional data
Keywords: methods: numerical - celestial mechanics - space vehicles
Abstract:
This paper introduces a new approach for solving the Kepler equation
for hyperbolic orbits. We provide here the Hyperbolic Kepler
Equation-Space Dynamics Group (HKE-SDG), a code to solve the equation.
Instead of looking for new algorithms, in this paper we have tried to
substantially improve well-known classic schemes based on the
excellent properties of the Newton-Raphson iterative methods. The key
point is the seed from which the iteration of the Newton-Raphson
methods begin. If this initial seed is close to the solution sought,
the Newton-Raphson methods exhibit an excellent behavior. For each one
of the resulting intervals of the discretized domain of the hyperbolic
anomaly a fifth degree interpolating polynomial is introduced, with
the exception of the last one where an asymptotic expansion is
defined. This way the accuracy of initial seed is optimized. The
polynomials have six coefficients which are obtained by imposing six
conditions at both ends of the corresponding interval: the polynomial
and the real function to be approximated have equal values at each of
the two ends of the interval and identical relations are imposed for
the two first derivatives. A different approach is used in the
singular corner of the Kepler equation - |M|<0.15 and 1<e<1.25 -
where an asymptotic expansion is developed.
In all simulations carried out to check the algorithm, the seed
generated leads to reach machine error accuracy with a maximum of
three iterations (∼99.8% of cases with one or two iterations) when
using different Newton-Raphson methods in double and quadruple
precision. The final algorithm is very reliable and slightly faster in
double precision (∼0.3 seconds). The numerical results confirm the use
of only one asymptotic expansion in the whole domain of the singular
corner as well as the reliability and stability of the HKE-SDG. In
double and quadruple precision it provides the most precise solution
compared with other methods.
Description:
The Hyperbolic Kepler Equation-Space Dynamics Group (HKE-SDG) code
solves the Kepler equation for hyperbolic keplerian orbits. The Maple
symbolic manipulator has been chosen to obtain the code in the C
programming language. The code is collected in the following modules:
main.c, keplerh.c, polinomios25.c polinomios25Q.c and hyperk.h. A zip
file includes all these modules together with readMe.pdf and
readMe.txt files in order to provide the instructions to compile and
run the HKE-SDG code. Users can compute the solution in double
precision by using the Modified Newton-Rhapson method.
File Summary:
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FileName Lrecl Records Explanations
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ReadMe 80 . This file
readme.pdf 512 165 ReadMe of the code
HKE-SDG_code.tar 512 800 All modules of the HKE-SDG code
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Acknowledgements:
Virginia Raposo-Pulido, v.raposo.pulido(at)upm.es
Jesus Pelaez, j.pelaez(at)upm.es
(End) Patricia Vannier [CDS] 02-Oct-2018