Bounds on the solution to a universal Kepler's equation

Prussing, John E.

doi:10.2514/6.1978-1405

Cited by 1 publication

(1 citation statement)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…While there are many articles discussing solutions for the elliptic case (see [1], [3], [4], [8], [9], [10], [13], among others), the hyperbolic case has received less attention. Prussing [11] and Serafin [12] gave upper bounds for the actual solution of the hyperbolic Kepler's equation, which can be used as starters for Newton's method since f g,L ≥ 0. Gooding and Odell [6] solved the hyperbolic Kepler's equation by using Newton's method starting from a well-tuned formula depending on the parameters g and L. Their approach gives a relative accuracy of 10 −20 with only two iterations.…”

Section: Introductionmentioning

confidence: 99%

Approximate solutions of the hyperbolic Kepler equation

Avendaño¹,

Martín-Molina²,

Ortigas-Galindo³

2015

Celest Mech Dyn Astr

View full text Add to dashboard Cite

We provide an approximate zero S(g, L) for the hyperbolic Kepler's equation S − g arcsinh(S) − L = 0 for g ∈ (0, 1) and L ∈ [0, ∞). We prove, by using Smale's α-theory, that Newton's method starting at our approximate zero produces a sequence that converges to the actual solution S(g, L) at quadratic speed, i.e. if S n is the value obtained after n iterations, then |S n − S| ≤ 0.5 2 n −1 | S − S|. The approximate zero S(g, L) is a piecewisedefined function involving several linear expressions and one with cubic and square roots. In bounded regions of (0, 1) × [0, ∞) that exclude a small neighborhood of g = 1, L = 0, we also provide a method to construct simpler starters involving only constants.

show abstract