Algorithms for Simultaneous Pad\'e Approximations

TL;DR

This paper introduces efficient algorithms for solving simultaneous Padé approximation problems over power series rings, leveraging recent advances in computational algebra to improve performance and solution completeness.

Contribution

It presents two novel algorithms for simultaneous Padé approximations that produce complete solution sets using advanced minimal approximant bases techniques.

Findings

Algorithms operate in near-optimal complexity $O(n^{\omega - 1} d)$.

Both algorithms produce reduced sub-bases for the solution set.

Results rely on recent breakthroughs in algebraic computation methods.

Abstract

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.