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This article is cited in 9 scientific papers (total in 9 papers)
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 $[f_0,\dots,f_m]$, $m\geqslant1$, 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 $(k,k,k,\dots,k,k)$, $(k+1,k,k,\dots,k,k)$, $(k+1,k+1,k,\dots,k,k)$, $\dots$, $(k+1,k+1,k+1,\dots,k+1,k)$ are already known at the point when the algorithm produces the Hermite-Padé polynomials corresponding to the multi-index $(k+1,k+1,k+1,\dots,k+1,k+1)$.
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.
Received: 17.03.2020 and 01.06.2021
Citation:
N. R. Ikonomov, S. P. Suetin, “A Viskovatov algorithm for Hermite-Padé polynomials”, Sb. Math., 212:9 (2021), 1279–1303
Linking options:
https://www.mathnet.ru/eng/sm9410https://doi.org/10.1070/SM9410 https://www.mathnet.ru/eng/sm/v212/i9/p94
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Abstract page: | 478 | Russian version PDF: | 45 | English version PDF: | 26 | Russian version HTML: | 116 | References: | 36 | First page: | 7 |
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