Fast computation of approximant bases in canonical form

TL;DR

This paper introduces faster algorithms for computing shifted Popov approximant bases in canonical form, extending previous methods to arbitrary shifts with improved efficiency and broadening applicability to problems like Hermite-Pade approximation.

Contribution

It extends existing algorithms to handle arbitrary shifts efficiently and modifies them to produce shifted Popov bases without increasing computational cost.

Findings

Achieves faster computation for arbitrary shifts in approximant bases.

Modifies existing algorithms to output shifted Popov bases directly.

Provides a new divide and conquer approach reducing general cases to known efficient cases.

Abstract

In this article, we design fast algorithms for the computation of approximant bases in shifted Popov normal form. We first recall the algorithm known as PM-Basis, which will be our second fundamental engine after polynomial matrix multiplication: most other fast approximant basis algorithms basically aim at efficiently reducing the input instance to instances for which PM-Basis is fast. Such reductions usually involve partial linearization techniques due to Storjohann, which have the effect of balancing the degrees and dimensions in the manipulated matrices. Following these ideas, Zhou and Labahn gave two algorithms which are faster than PM-Basis for important cases including Hermite-Pade approximation, yet only for shifts whose values are concentrated around the minimum or the maximum value. The three mentioned algorithms were designed for balanced orders and compute approximant bases that are generally not normalized. Here, we show how they can be modified to return the shifted Popov basis without impact on their cost bound; besides, we extend Zhou and Labahn's algorithms to arbitrary orders. Furthermore, we give an algorithm which handles arbitrary shifts with one extra logarithmic factor in the cost bound compared to the above algorithms. To the best of our knowledge, this improves upon previously known algorithms for arbitrary shifts, including for particular cases such as Hermite-Pade approximation. This algorithm is based on a recent divide and conquer approach which reduces the general case to the case where information on the output degree is available. As outlined above, we solve the latter case via partial linearizations and PM-Basis.