  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.
