  Let $F(x)= \sum_{\nu\in\NN^d} F_\nu x^\nu$ be a multivariate power series
with complex coefficients that converges in a neighborhood of the origin.
Assume $F=G/H$ for some functions $G$ and $H$ holomorphic in a neighborhood of
the origin. We derive asymptotics for the coefficients $F_{r\alpha}$ as $r \to
\infty$ with $r\alpha \in \NN^d$ for $\alpha$ in a permissible subset of
$d$-tuples of positive reals. More specifically, we give an algorithm for
computing arbitrary terms of the asymptotic expansion for $F_{r\alpha}$ when
the asymptotics are controlled by a transverse multiple point of the analytic
variety $H = 0$. This improves upon earlier work by R. Pemantle and M. C.
Wilson.
  We have implemented our algorithm in Sage and apply it to obtain accurate
numerical results for several rational combinatorial generating functions.
