## Abstract

We describe how to compute simultaneous Hermite–Padé approximations, over a polynomial ring K[x] for a field K using O^{∼}(n^{ω−1}td) operations in K, where d is the sought precision, where n is the number of simultaneous approximations using t<n polynomials, and where O(n^{ω}) is the cost of multiplying n×n matrices over K. We develop two algorithms using different approaches. Both algorithms return a reduced sub-basis that generates the complete set of solutions to the input approximation problem that satisfy the given degree constraints. Previously, the cost O^{∼}(n^{ω−1}td) has only been reached with randomized algorithms finding a single solution for the case t<n. Our results are made possible by recent breakthroughs in fast computations of minimal approximant bases and Hermite–Padé approximations for the case t≥n.

Original language | English |
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Journal | Journal of Symbolic Computation |

Volume | 102 |

Pages (from-to) | 279-303 |

ISSN | 0747-7171 |

DOIs | |

Publication status | Published - 1 Jan 2020 |

## Keywords

- Approximant bases
- Padé approximation
- Structured linear systems