A Viskovatov algorithm for Hermite-Pad polynomials,Sbornik: Mathematics

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A Viskovatov algorithm for Hermite-Pad polynomials
Sbornik: Mathematics ( IF 0.8 ) Pub Date : 2021-11-15 , DOI: 10.1070/sm9410
N. R. Ikonomov ₁ , S. P. Suetin ₂

Affiliation

We propose and justify an algorithm for producing Hermite- Pad polynomials of type I for an arbitrary tuple of $m+1$ formal power series $[f_0,\dots,f_m]$ , $m\geq1$ , about the point $z=0$ ( $f_j\in\mathbb{C}[[z]]$ ) under the assumption that the series have a certain (‘general position’) nondegeneracy property. This algorithm is a straightforward extension of the classical Viskovatov algorithm for constructing Pad polynomials (for $m=1$ our algorithm coincides with the Viskovatov algorithm).

The algorithm is based on a recurrence relation and has the following feature: all the Hermite-Pad polynomials corresponding to the multi- indices $(k,k,k,\dots,k,k)$ , $(k+1,k,k,\dots,k,k)$ , $(k+1,k+1,k,\dots,k,k)$ , $\dots$ , $(k+1,k+1,k+1,\dots,k+1,k)$ are already known at the point when the algorithm produces the Hermite-Pad polynomials corresponding to the multi- index $(k+1,k+1,k+1,\dots,k+1,k+1)$ .

We show how the Hermite-Pad polynomials corresponding to different multi-indices can be found recursively via this algorithm by changing the initial conditions appropriately.

At every step $n$ , the algorithm can be parallelized in $m+1$ independent evaluations.

Bibliography: 30 titles.

中文翻译：

Hermite-Pad 多项式的 Viskovatov 算法

我们提出并用于生产I型Hermite-垫多项式的任意元组证明的算法 $m+1$ 形式幂级数 $[f_0,\dots,f_m]$ ， $m\geq1$ 大约点 $z=0$ （ $f_j\in\mathbb{C}[[z]]$ 假设该系列有一定的（“总的立场”）非退化属性下）。该算法是经典 Viskovatov 算法的直接扩展，用于构造 Pad 多项式（因为 $m=1$ 我们的算法与 Viskovatov 算法一致）。

该算法基于递推关系，具有以下特点：当算法产生多个索引对应的 Hermite-Pad 多项式时，所有对应于多个索引的 Hermite-Pad 多项式 $(k,k,k,\dots,k,k)$ ， $(k+1,k,k,\dots,k,k)$ , $(k+1,k+1,k,\dots,k,k)$ , $\点$ , $(k+1,k+1,k+1,\dots,k+1,k)$ 都是已知的。索引 $(k+1,k+1,k+1,\dots,k+1,k+1)$ .

我们展示了如何通过该算法通过适当地改变初始条件来递归地找到对应于不同多指数的 Hermite-Pad 多项式。

在每一步 $n$ ，算法都可以在 $m+1$ 独立评估中并行化。

参考书目：30个标题。

更新日期：2021-11-15

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