### Algorithms for Simultaneous Padé Approximations

**2016-02-02**

1602.00836 | cs.SC

We describe how to solve simultaneous Pad\'e approximations over a power
series ring $K[[x]]$ for a field $K$ using $O~(n^{\omega - 1} d)$ operations in
$K$, where $d$ is the sought precision and $n$ is the number of power series to
approximate. We develop two algorithms using different approaches. Both
algorithms return a reduced sub-bases that generates the complete set of
solutions to the input approximations problem that satisfy the given degree
constraints. Our results are made possible by recent breakthroughs in fast
computations of minimal approximant bases and Hermite Pad\'e approximations.

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