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

TL;DR

This paper introduces an approximate zero for the hyperbolic Kepler's equation, proving quadratic convergence of Newton's method from this initial guess, with explicit formulas and simplified starters for most parameter regions.

Contribution

The authors develop a new approximate zero for the hyperbolic Kepler's equation and prove its effectiveness using Smale's alpha-theory, including explicit formulas and simplified starters.

Findings

Newton's method converges quadratically from the approximate zero.

Explicit piecewise formulas for the approximate zero are provided.

Simpler starters are constructed for most parameter regions.

Abstract

We provide an approximate zero $\widetilde{S}(g,L)$ for the hyperbolic Kepler's equation $S-g\, {\rm asinh} (S)-L=0$ 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 $|S_n-S|\leq 0.5^{2^n-1}|\widetilde{S}-S|$. The approximate zero $\widetilde{S}(g,L)$ 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.