scholarly journals Fast Switch and Spline Function Inversion Algorithm with Multistep Optimization and k-Vector Search for Solving Kepler’s Equation in Celestial Mechanics

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2017
Author(s):  
Daniele Tommasini ◽  
David N. Olivieri

Obtaining the inverse of a nonlinear monotonic function f(x) over a given interval is a common problem in pure and applied mathematics, the most famous example being Kepler’s description of orbital motion in the two-body approximation. In traditional numerical approaches, this problem is reduced to solving the nonlinear equation f(x)−y=0 in each point y of the co-domain. However, modern applications of orbital mechanics for Kepler’s equation, especially in many-body problems, require highly optimized numerical performance. Ongoing efforts continually attempt to improve such performance. Recently, we introduced a novel method for computing the inverse of a one-dimensional function, called the fast switch and spline inversion (FSSI) algorithm. It works by obtaining an accurate interpolation of the inverse function f−1(y) over an entire interval with a very small generation time. Here, we describe two significant improvements with respect to the performance of the original algorithm. First, the indices of the intervals for building the spline are obtained by k-vector search combined with bisection, thereby making the generation time even smaller. Second, in the case of Kepler’s equation, a multistep method for the optimized calculation of the breakpoints of the spline polynomial was designed and implemented in Cython. We demonstrate results that accurately solve Kepler’s equation for any value of the eccentricity e∈[0,1−ϵ], with ϵ=2.22×10−16, which is the limiting error in double precision. Even with modest current hardware, the CPU generation time for obtaining the solution with high accuracy in a large number of points of the co-domain can be kept to around a few nanoseconds per point.

2018 ◽  
Vol 619 ◽  
pp. A128 ◽  
Author(s):  
M. Zechmeister

Context. Many algorithms to solve Kepler’s equations require the evaluation of trigonometric or root functions. Aims. We present an algorithm to compute the eccentric anomaly and even its cosine and sine terms without usage of other transcendental functions at run-time. With slight modifications it is also applicable for the hyperbolic case. Methods. Based on the idea of CORDIC, our method requires only additions and multiplications and a short table. The table is independent of eccentricity and can be hardcoded. Its length depends on the desired precision. Results. The code is short. The convergence is linear for all mean anomalies and eccentricities e (including e = 1). As a stand-alone algorithm, single and double precision is obtained with 29 and 55 iterations, respectively. Half or two-thirds of the iterations can be saved in combination with Newton’s or Halley’s method at the cost of one division.


2020 ◽  
Vol 64 (12) ◽  
pp. 1060-1066
Author(s):  
M. K. Abubekerov ◽  
N. Yu. Gostev

1983 ◽  
Vol 31 (3) ◽  
pp. 317-328 ◽  
Author(s):  
T. M. Burkardt ◽  
J. M. A. Danby

1997 ◽  
Vol 51 (1) ◽  
pp. 59-65 ◽  
Author(s):  
Jacques Dutka

2010 ◽  
Vol 108 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Jeremy J. Davis ◽  
Daniele Mortari ◽  
Christian Bruccoleri

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