Abstract The solution of Kepler’s equation is accomplished via families of hybrid and table lookup techniques. A new arithmetic operation timing approach with general application to scientific programming is used to accurately estimate performance of each Kepler’s equation solution technique. The hybrid approaches couple a power series expansion starting approximation with nine different types of higher-order corrective step methods. The resulting computationally efficient non-iterative methods avoid “if” statements and directly yield in-plane Euler rotation angles necessary to map orbit elements to orbital position. The best-performing of the nine hybrid methods are up to two times faster than the original efficient Laguerre iterative method and achieve worst-case resultant true anomaly accuracies down to machine precision at 3 × 10−11° for eccentricities up to 0.999999. This matches or exceeds the performance of iterative methods and translates to less than three micro-meters at GEO altitude. Meanwhile, six digital approaches were explored, with the best table lookup approach boasting a ten-fold speed increase with a worst-case accuracy of 1 × 10−6° (154 mm) for an eccentricity of 0.999999. These combinations of accuracy and speed performance make both the hybrid and table lookup approaches well-suited for in-line incorporation into a wide range of low- to high-fidelity semi-analytic orbit propagators and multi-threaded or vector programming languages and computing hardware (GPUs, etc.). And finally, a new operator-based computational timing technique is designed and employed to estimate the performance of math-laden computer programs, with sample application to all of the aforementioned Kepler’s equation solution techniques.
Efficient Solutions of Kepler’s Equation via Hybrid and Digital Approaches
The Journal of the Astronautical Sciences ; 62 , 4 ; 271-297
01.12.2015
27 pages
Aufsatz (Zeitschrift)
Elektronische Ressource
Englisch
Cost of computational operators , Laguerre , Table lookup , Taylor series , Halley higher-order methods , Nonlinear bivariate interpolation , Kepler’s equation , Memory mapping Engineering , Aerospace Technology and Astronautics , Space Sciences (including Extraterrestrial Physics, Space Exploration and Astronautics) , Mathematical Applications in the Physical Sciences
Efficient Solutions of Kepler’s Equation via Hybrid and Digital Approaches
Online Contents | 2015
|Efficient Solutions of Kepler’s Equation via Hybrid and Digital Approaches
Online Contents | 2015
|Erratum to: Efficient Solutions of Kepler’s Equation via Hybrid and Digital Approaches
Online Contents | 2017
|Erratum to: Efficient Solutions of Kepler’s Equation via Hybrid and Digital Approaches
Online Contents | 2016
|Erratum to: Efficient Solutions of Kepler’s Equation via Hybrid and Digital Approaches
Springer Verlag | 2016
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