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 Mat. Sb., 2021, Volume 212, Number 9, Pages 94–118 (Mi msb9410)

A Viskovatov algorithm for Hermite-Padé polynomials

N. R. Ikonomova, S. P. Suetinb*

a Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: We propose and justify an algorithm for producing Hermite-Padé polynomials of type I for an arbitrary tuple of $m+1$ formal power series $[f_0,…,f_m]$, $m\geq1$, 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 Padé 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-Padé polynomials corresponding to the multi-indices $(k,k,k,…,k,k)$, $(k+1,k,k,…,k,k)$, $(k+1,k+1,k,…,k,k)$, …, $(k+1,k+1,k+1,…,k+1,k)$ are already known at the point when the algorithm produces the Hermite-Padé polynomials corresponding to the multi-index $(k+1,k+1,k+1,…,k+1,k+1)$.
We show how the Hermite-Padé 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.

Keywords: formal power series, Hermite-Padé polynomials, Viskovatov algorithm.

 Funding Agency Grant Number Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1614 The research of S. P. Suetin was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614).

* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm9410

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English version:
Sbornik: Mathematics, 2021, 212:9, 1279–1303

UDC: 517.53
MSC: 41A21

Citation: N. R. Ikonomov, S. P. Suetin, “A Viskovatov algorithm for Hermite-Padé polynomials”, Mat. Sb., 212:9 (2021), 94–118; Sb. Math., 212:9 (2021), 1279–1303

Citation in format AMSBIB
\Bibitem{IkoSue21} \by N.~R.~Ikonomov, S.~P.~Suetin \paper A Viskovatov algorithm for Hermite-Pad\'e polynomials \jour Mat. Sb. \yr 2021 \vol 212 \issue 9 \pages 94--118 \mathnet{http://mi.mathnet.ru/msb9410} \crossref{https://doi.org/10.4213/sm9410} \transl \jour Sb. Math. \yr 2021 \vol 212 \issue 9 \pages 1279--1303 \crossref{https://doi.org/10.1070/SM9410}