Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Solving Kepler's equation via Smale's $\alpha$-theory   We obtain an approximate solution $\tilde{E}=\tilde{E}(e,M)$ of Kepler's
equation $E-e\sin(E)=M$ for any $e\in[0,1)$ and $M\in[0,\pi]$. Our solution is
guaranteed, via Smale's $\alpha$-theory, to converge to the actual solution $E$
through Newton's method at quadratic speed, i.e. the $n$-th iteration produces
a value $E_n$ such that $|E_n-E|\leq (\frac12)^{2^n-1}|\tilde{E}-E|$. The
formula provided for $\tilde{E}$ is a piecewise rational function with
conditions defined by polynomial inequalities, except for a small region near
$e=1$ and $M=0$, where a single cubic root is used. We also show that the root
operation is unavoidable, by proving that no approximate solution can be
computed in the entire region $[0,1)\times[0,\pi]$ if only rational functions
are allowed in each branch.

32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
11
The categories of this paper are: hep-lat