A Viskovatov algorithm for Hermite-Pad\'e polynomials

N. R. Ikonomov ; S. P. Suetin

Sbornik. Mathematics, Tome 212 (2021) no. 9, pp. 1279-1303

Résumé

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.

Détail

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