标准型近似基的快速计算
Fast computation of approximant bases in canonical form
摘要：
在这篇文章中，我们设计了快速算法的移动Popov范式的近似基的计算。我们首先回顾称为PM-Basis的算法，这将是我们在多项式矩阵乘法之后的第二个基本引擎：大多数其他快速近似基算法基本上旨在有效地将输入实例减少到PM-Basis快速的实例。这样的减少通常涉及部分线性化技术，由于Storjohann，它具有平衡的程度和尺寸的操纵矩阵的效果。根据这些想法，周和Labahn给出了两个算法，它们在包括Hermite-Pade近似在内的重要情况下比PM-基更快，但仅适用于其值集中在最小值或最大值周围的移位。上述三种算法是为平衡阶数而设计的，并计算通常不归一化的近似基。在这里，我们展示了如何修改它们以返回移位Popov基而不影响它们的成本界;此外，我们将Zhou和Labahn的算法扩展到任意阶。此外，我们给出了一个算法，处理任意移位的成本界相比，上述算法的一个额外的对数因子。据我们所知，这改进了以前已知的算法，用于任意移位，包括特定情况下，如厄米-帕德近似。该算法是基于最近的分而治之的方法，减少了一般情况下的输出度的信息是可用的。如上所述，我们通过部分线性化和PM基解决了后一种情况。
英文摘要：
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.
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