Approximate solutions of the hyperbolic Kepler equation
Abstract
We provide an approximate zero widetilde{S}(g,L) for the hyperbolic Kepler's equation Sg {{arcsinh}}(S)L=0 for gin (0,1) and Lin [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_nS≤ 0.5^{2^n1}widetilde{S}S. The approximate zero widetilde{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.
 Publication:

Celestial Mechanics and Dynamical Astronomy
 Pub Date:
 December 2015
 DOI:
 10.1007/s1056901596450
 arXiv:
 arXiv:1503.01641
 Bibcode:
 2015CeMDA.123..435A
 Keywords:

 Physics  Classical Physics;
 Mathematical Physics;
 Mathematics  Numerical Analysis;
 Physics  Computational Physics
 EPrint:
 14 pages, 2 figures