Improved Initial Guesses for Numerical Solutions of Kepler's Equation

Kevin J Napier

REVIEW 2 major objections 3 minor 4 references

Improved Initial Guesses for Numerical Solutions of Kepler's Equation

T0 review · 2 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Symbolic regression discovers simple initial guesses that make iterative Kepler solvers faster, with the largest gains for hyperbolic orbits.

desk verdict Good elliptic initial guess, but the hyperbolic speedup is only shown in a small-mean-anomaly box and likely reverses at large M. read the letter →

arxiv 2411.15374 v1 pith:OLW5ARQ4 submitted 2024-11-22 astro-ph.EP astro-ph.IMphysics.comp-ph

Kevin J Napier This is my paper · ORCID

classification astro-ph.EPastro-ph.IMphysics.comp-ph MSC 65H05

keywords Kepler'sequationinitialguesssymbolicregressiongeneticalgorithmellipticalorbithyperbolicnumericalrootfindingcelestialmechanics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Kepler's equation relates mean anomaly to eccentric anomaly and must be solved numerically billions of times in orbit simulations. This paper argues that the standard starting guesses for iterative solvers can be improved cheaply: a machine-learning search over simple arithmetic expressions produced the elliptical guess $E = e\sin M + \max\{M, e(\sin M + 0.591)\}$ and the hyperbolic guess $E = \min\{M,e\}/(e - 0.538)$. In benchmarks with a quartic root-finding algorithm, the new guesses match or beat the canonical guesses, giving modest time savings on elliptical orbits and large savings on hyperbolic orbits. The practical payoff is that any orbit-propagation code can adopt these two one-line formulas without adding memory lookups or branching.

What carries the argument

The load-bearing object is the initial-guess expression itself, produced by symbolic regression with a parsimony penalty. The algorithm explores the space of short formulas built from $e$, $M$, $\sin M$, and constants of order unity using only $+$, $-$, $\times$, $\min$, and $\max$, so the result is a closed-form, branch-free expression that costs almost nothing to evaluate. The crucial structural trick is the $\max$ operation in the elliptical guess: it partitions the $(e,M)$ plane into two regions, applying the cheap canonical guess where it is already good and a corrected formula only where it is needed. For the hyperbolic guess, the operation $\min\{M,e\}$ plays a similar role, approximating the behavior of the true solution as a function of whichever of $M$ and $e$ is limiting. The parsimony penalty is what keeps the expressions short enough to be evaluated faster than the extra iterations they save.

What would settle it

Benchmark both the new and canonical guesses using Newton's method and a secant method in a compiled language such as C or Fortran on a grid of $e\in[0.9,1)$ and $M\in[0,\pi/4]$ for the elliptical case, and $e\in(1,4]$, $M\in[0,\pi]$ for the hyperbolic case. If the new guesses do not reduce the median iteration count or wall-clock time to reach $10^{-15}$ accuracy in these settings, then the paper's central claim of a general speed improvement is falsified. A simpler check: verify that for hyperbolic orbits with $e$ just above 1 and small $M$, the new guess keeps the Newton iteration inside the basin of convergence; if it occasionally diverges, the speed gain is not real.

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Extended reading notes

Core claim

The paper's central claim is that a genetic symbolic-regression search, restricted to the operations $+$, $-$, $\times$, $\min$, $\max$ and the variables $e$, $M$, $\sin M$, yields initial guesses that outperform the standard textbook guesses for both forms of Kepler's equation. For elliptical orbits the discovered guess coincides with the classic $M + e\sin M$ in most of parameter space but uses a $\max$ to switch to a better fit in the high-eccentricity, near-pericenter region, where the canonical guess is poor. For hyperbolic orbits the discovered guess $\min\{M,e\}/(e-0.538)$ beats the usual $\ln(2M/e+1.8)$ over most of the domain and appears to improve even at arbitrarily large eccentricity. The speed advantage is measured in iteration counts and wall-clock time at $10^{-15}$ (elliptical) and $10^{-14}$ (hyperbolic) precision. The paper presents these as practical improvements, not a theoretical ceiling: better guesses may exist, and the technique is offered as a template for finding them.

Load-bearing premise

The claimed speedups rest on timing measurements from a single, not fully specified quartic root-finding implementation written in Rust; if that solver, compiler, or machine is not representative of how most codes solve Kepler's equation, the improvements may shrink or vanish in practice.

Editorial extensions

If this is right

Any celestial-mechanics code that solves Kepler's equation by iteration can replace the canonical initial guesses with Equations (2) and (4) and expect equal or faster convergence, with the largest gains for high-eccentricity elliptical orbits near pericenter and for hyperbolic orbits.
The hyperbolic guess $\min\{M,e\}/(e-0.538)$ remains effective even as eccentricity grows without bound, making it relevant for unbound trajectories and interstellar objects.
Because the new guesses are plain analytic expressions, they carry no memory footprint and cause no cache misses, unlike grid-interpolation or neural-network initializers.
The symbolic-regression recipe can be rerun with custom cost weightings, such as emphasizing rare high-eccentricity orbits, to tailor initial guesses to specific applications.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The machine-dependent $\sin$ call in the elliptical guess could likely be replaced by a truncated Taylor series; the author flags this as a possibility but did not implement it, and a direct benchmark would show whether the trade-off is worthwhile.
The same genetic search over cheap arithmetic expressions could be applied to other transcendental equations in orbital mechanics, such as the Barker equation or universal-variable formulation, where analogous speedups may exist.
In very large simulations that solve Kepler's equation millions of times, even one saved iteration per solve can translate into substantial wall-clock savings; the paper's benchmark does not directly quantify this bulk-throughput effect.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the elliptic initial guess (Eq. 2) is a genuine, modest win, and the paper is transparent about how the constants were found. But the headline hyperbolic claim is over-sold: Eq. (4) caps at min{M,e} and is bad for M >> e, a regime the benchmarks don't cover. The paper deserves a referee, but not as is.

The genuinely new thing is the use of symbolic regression to fit initial guesses, yielding two closed-form expressions that are cheaper than Markley's and better than the classic E = M + e sin M near pericenter for high e. That part is plausible and useful. The author also deserves credit for noting the search recovered canonical guesses under parsimony, which is a nice sanity check.

The soft spots. First, the hyperbolic formula. For fixed e>1 and M >> e, E0 = e/(e-0.538) is roughly constant while true E grows as ln(2M/e). The benchmark only explores M up to 3π/4, so the claimed 'major speed improvements for hyperbolic orbits' does not extend to large M. A root finder starting at ~1.5 instead of ~5 will need extra iterations. That's a real limitation, not a quibble. The abstract and conclusions should be qualified, or the formula should be patched with a log term for large M. Second, the benchmarks rely on an unspecified 'quartic root-finding algorithm' in Rust, no code or data are released. That makes the speed claims hard to audit, though the iteration counts are at least reproducible in principle from the formulas. The elliptical formula is the more solid contribution; the hyperbolic one is promising in a limited domain.

Bottom line: this is a useful, honest little paper about a computational subroutine. It isn't a field changer, and it doesn't need to be. A serious referee should see it, primarily to push the author to fix the hyperbolic claim and release benchmarks. I would not cite it in my own work until the large-M behavior is addressed. Reading-group material? Maybe, as a cautionary example of extrapolating from a limited parameter scan.

Referee Report

2 major / 3 minor

Summary. The paper uses symbolic regression with a genetic algorithm to derive two new initial guesses for iterative solvers of Kepler's equation: Eq. (2) for elliptical orbits, E = e sin M + max{M, e(sin M + 0.591)}, and Eq. (4) for hyperbolic orbits, E = min{M,e}/(e - 0.538). Benchmarks using a quartic root-finding algorithm in Rust (Figs. 1 and 2) show modest speed improvements for elliptical orbits and larger speed improvements for hyperbolic orbits within the scanned parameter ranges. The author concludes that the new guesses improve on canonical initial guesses and that the symbolic-regression approach yields interpretable, low-cost expressions.

Significance. If the reported speedups hold, the results are practically useful for celestial mechanics software that solves Kepler's equation in high-throughput settings. The proposed guesses are simple, interpretable, and cheap to evaluate, and the paper is transparent that the constants 0.591 and 0.538 were found by fitting rather than derived, which avoids circularity in the benchmark comparison. The external comparison against standard guesses is a fair and relevant test of the central speed claim. However, the absence of released code and the incomplete specification of the root-finding algorithm weaken the reproducibility of the benchmark, and the hyperbolic guess is shown here to fail in a physically relevant large-mean-anomaly regime that the paper does not mention.

major comments (2)

[Section 4, Eq. (4)] The hyperbolic initial guess is not uniformly better outside the benchmarked domain. For M >= e, Eq. (4) reduces to the constant e/(e - 0.538), independent of M, whereas the true hyperbolic anomaly grows logarithmically, E ~ ln(2M/e). For example, with e = 1.5 and M = 100, Eq. (4) gives E0 = 1.56, the canonical Danby guess gives about 4.91, and the true solution is about 4.94; a iterative solver started at 1.56 will require many more iterations than one started at 4.91. Since Figure 2 only scans M in [0, 3*pi/4] and e in [1, 4], the claimed 'major speed improvements for hyperbolic orbits' is not supported for larger M and in fact reverses. Please either restrict the claim to the benchmarked range or modify the guess to include an asymptotic regime, e.g., a switch to a logarithmic form for large M.
[Section 3, footnote 3] The benchmarks rely on an unspecified 'quartic root-finding algorithm'; no source code or data are released. The central claim of this paper is a speed improvement, and that claim necessarily depends on the solver's update rule, stopping criterion, and implementation. Without stating which algorithm is used (or giving a reference) and ideally releasing the benchmark code, the results cannot be independently checked. Please specify the algorithm, report iteration counts and timing with error bars, and consider making the code available.

minor comments (3)

[Section 3, Figures 1 and 2] The timing panels show point values without error bars or confidence intervals; since criterion.rs can provide such intervals, please include them to quantify the 'some noise' mentioned in the hyperbolical figure caption.
[Section 2, paragraph 2] The details of the symbolic regression and genetic algorithm (fitness function, parsimony penalty strength, population size, number of generations) are not given, so the search procedure is not reproducible; adding a short description or reference would strengthen the methodological contribution.
[Section 4, Eq. (4)] As written, Eq. (4) has a singular denominator at e = 0.538. For hyperbolic orbits e > 1, so this is not a practical issue, but please state the domain of validity explicitly in the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the initial guesses are transparently fitted expressions, and the speed claims rest on external benchmarks against canonical guesses.

full rationale

The paper's central content is a symbolic-regression search over analytic expressions, and it transparently presents the resulting constants as fitted quantities: the algorithm 'converged on' Eq. (2) and Eq. (4), with no claim that they are derived from first principles. The improvement claim is then tested by an external criterion: wall-clock time and iteration counts of a quartic root-finding algorithm, compared against the canonical guesses from Danby. That benchmark is not forced by the fitting objective, because the fit minimizes residual error over the parameter space, while the speed comparison is a separate empirical measurement. No load-bearing self-citation chain appears: the cited references are Danby, Markley, Philcox et al., and a benchmarking library, none of which are prior work by the author that smuggles in the result. The hyperbolic guess's bounded behavior at large M, noted by a skeptical reading, is a correctness/extrapolation concern outside the benchmarked domain, not a circularity: the formula is visibly fitted to the small-M search region, and the paper does not present it as an asymptotic derivation. Thus the derivation chain is self-contained and the reported predictions are not equivalent to the inputs by construction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper's contribution rests on two numerically fitted constants (0.591 and 0.538) plus standard assumptions about the domain and the root-finding setup. There is no new physical entity or first-principles derivation. The benchmark conclusions depend on an unspecified solver and a particular Rust measurement environment.

free parameters (2)

c_ellip = 0.591 = 0.591
Fitted constant in the elliptical initial guess, Eq. (2). The symbolic regressor was allowed scalar quantities of order unity, and this value was selected to minimize initial-guess error over the parameter space.
c_hyper = 0.538 = 0.538
Fitted constant in the hyperbolic initial guess, Eq. (4). Found by symbolic regression using scalars of order unity; no analytic derivation is given.

assumptions (5)

standard math Kepler's equation is transcendental and must be solved numerically with iterative root finding.
This is the standard mathematical status of Kepler's equation, stated in the introduction (Section 1).
domain assumption It is sufficient to solve for M in [0, pi] by exploiting symmetry.
The paper restricts the parameter space using symmetry in M (Section 2, footnote 1). If this symmetry is not valid for all real usage, the fitted guess may not apply where it is needed.
domain assumption Uniform sampling of the parameter space during fitting is representative of actual orbit populations.
The symbolic regressor was fit on a uniform sample of e and M (Section 2). Speed improvements are benchmarked on the same kind of uniform grid, so the reported gains may not hold for non-uniform real-world usage distributions.
domain assumption The quartic root-finding algorithm used in benchmarks converges for all test cases.
The benchmarks rely on an unspecified quartic root-finding algorithm (Section 3). The speed comparison would be meaningless if that algorithm failed on some initial guesses or if the chosen root-finder is atypical.
domain assumption The sin M implementation in the chosen math library gives acceptable accuracy for the initial guess.
The author notes that the use of sin M makes the initial guess machine dependent (Section 5). The fitted constants assume a standard sin implementation.

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Cite this review

Pith. "Pith review of Improved Initial Guesses for Numerical Solutions of Kepler's Equation." pith.science (2026). https://pith.science/paper/OLW5ARQ4

@misc{pith2026241115374,
  author       = {Pith},
  title        = {Pith review of: Improved Initial Guesses for Numerical Solutions of Kepler's Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OLW5ARQ4}},
  note         = {Machine review of arXiv:2411.15374}
}

read the original abstract

Numerical solutions of Kepler's Equation are critical components of celestial mechanics software, and are often computation hot spots. This work uses symbolic regression and a genetic learning algorithm to find new initial guesses for iterative Kepler solvers for both elliptical and hyperbolic orbits. The new initial guesses are simple to implement, and result in modest speed improvements for elliptical orbits, and major speed improvements for hyperbolic orbits.

Figures

Figures reproduced from arXiv: 2411.15374 by the authors.

**Figure 1.** — (Top) Average time taken to converge on a solution to Kepler’s Equation at a precision of 10−15, parameterized as a function of e and M. The new initial guess (left) results in faster convergence than the canonical initial guess (right) for high-eccentricity orbits near pericenter. The performance is similar in the rest of the parameter space. (Bottom) Number of iterations taken to converge on a solution to Kepler… view at source ↗

**Figure 2.** — (Top) Average time taken to converge on a solution to Kepler’s Equation at a precision of 10−14, parameterized as a function of e and M. There is some noise in these benchmarks, but it is clear that the new initial guess (left) results in faster convergence than the canonical initial guess (right) for most of the parameter space. (Bottom) Number of iterations taken to converge on a solution to Kepler’s Equation at… view at source ↗

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Reference graph

Works this paper leans on

4 extracted references · 2 canonical work pages

[1]

2014–2024, criterion.rs: Statistics-driven micro-benchmarking library for Rust

Aparicio, J., & Heisler, B. 2014–2024, criterion.rs: Statistics-driven micro-benchmarking library for Rust. https://github.com/japaric/criterion.rs

work page 2014
[2]

Danby, J. M. A. 1992, Fundamentals of Celestial Mechanics (Richmond, V A: Willmann-Bell, Inc.)

work page 1992
[3]

Markley, F. L. 1995, Celestial Mechanics and Dynamical Astronomy, 63, 101, doi: 10.1007/BF00691917

doi:10.1007/bf00691917 1995
[4]

Philcox, O. H. E., Goodman, J., & Slepian, Z. 2021, MNRAS, 506, 6111, doi: 10.1093/mnras/stab1296 This paper was built using the Open Journal of Astrophysics LATEX template. The OJA is a journal which provides fast and easy peer review for new papers in the astro-ph section of the arXiv, making the reviewing process simpler for authors and referees alike....

doi:10.1093/mnras/stab1296 2021

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

2014–2024, criterion.rs: Statistics-driven micro-benchmarking library for Rust

Aparicio, J., & Heisler, B. 2014–2024, criterion.rs: Statistics-driven micro-benchmarking library for Rust. https://github.com/japaric/criterion.rs

work page 2014

[2] [2]

Danby, J. M. A. 1992, Fundamentals of Celestial Mechanics (Richmond, V A: Willmann-Bell, Inc.)

work page 1992

[3] [3]

Markley, F. L. 1995, Celestial Mechanics and Dynamical Astronomy, 63, 101, doi: 10.1007/BF00691917

doi:10.1007/bf00691917 1995

[4] [4]

Philcox, O. H. E., Goodman, J., & Slepian, Z. 2021, MNRAS, 506, 6111, doi: 10.1093/mnras/stab1296 This paper was built using the Open Journal of Astrophysics LATEX template. The OJA is a journal which provides fast and easy peer review for new papers in the astro-ph section of the arXiv, making the reviewing process simpler for authors and referees alike....

doi:10.1093/mnras/stab1296 2021