Two fast and accurate routines for solving the elliptic Kepler equation for all values of the eccentricity and mean anomaly

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Table 1

Performance of the CNR solution of KE with different seeds.

With the PC seed E₀ of Eq. (11)

e
			(ns/sol.)
0.1	3.93	4	34
0.3	4.38	5	39
0.5	4.59	5	41
0.7	4.85	6	42
0.9	5.06	6	45
0.99	5.26	8	47

With the OG seed E₀ of Eq. (13)

e
			(ns/sol.)

0.1	3.85	4	30
0.3	4.08	5	35
0.5	4.28	5	36
0.7	4.58	5	37
0.9	4.75	6	43
0.99	4.65	6	43

With the CAL seed E₀ of Eq. (15)

e
			(ns/sol.)

0.1	3.42	4	26
0.3	3.77	4	30
0.5	3.92	5	32
0.7	4.14	5	33
0.9	4.24	6	34
0.99	4.29	8	35

With the rational seed E₀ of Eq. (19)

e
			(ns/sol.)

0.1	3.42	4	28
0.3	3.80	4	32
0.5	3.83	4	32
0.7	3.43	4	29
0.9	3.98	5	34
0.99	4.09	8	34

Notes. Performance of the CNR solution of KE with the classical stopping condition of Eq. (7) using four different seeds, when the accuracy is set to the value rad for M ∈ [0, 2π]. These results correspond to different values of the eccentricity e and to a homogeneous set of N = 10⁸ values of M ∈ [0, 2π]. The average and maximum numbers of CNR iterations (i.e., computations of Δ_n s), and , each corresponding to two transcendental function evaluations, are shown in the second and third column, respectively. The last column presents the average CPU execution time per solution, (in nanoseconds), obtained with an optimized Cython implementation of the CNR method employing multithreaded loops and executed on the hardware previously specified. For higher values of e, the CNR algorithm cannot attain the accuracy , and another routine, such as the ENRKE or the ENP5KE, which switch to the bisection method in the critical region, has to be used.

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