A new method for obtaining approximate solutions of the hyperbolic Kepler's equation

Abstract

We provide an approximate zero $\tilde{S}(g,L)$ for the hyperbolic Kepler's equation $S-g\mathrm{asinh}(S)-L=0$ for $g\in (0,1)$ and $L\in [0,\infty )$. We prove, by using Smale's $\alpha$-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 $|S_n-S|\le 0.5^{2^n-1}|\tilde{S}-S|$. The approximate zero $\tilde{S}(g,L)$ is a piecewise-defined function involving several linear expressions and one with cubic and square roots. In bounded regions of $(0,1)\times [0,\infty )$ that exclude a small neighborhood of $g=1,L=0$, we also provide a method to construct simpler starters involving only constants.

14 pages, 2 figures

References in corpus (1)

• Solving Kepler's equation via Smale's $\alpha$-theory

Cited by in corpus (2)

• Comment on "An efficient code to solve the Kepler equation. Elliptic case"
• Bivariate Infinite Series Solution of Kepler's Equations