Viskovatov algorithm for Hermite-Pad\'e polynomials

TL;DR

This paper introduces an extension of the classical Viskovatov algorithm to compute Hermite-Padé polynomials of type I for multiple formal power series, enabling efficient, parallelizable computation under certain nondegeneracy conditions.

Contribution

The paper presents a new recurrence-based algorithm for Hermite-Padé polynomials that generalizes the Viskovatov algorithm to multiple series and allows parallel computation.

Findings

Algorithm extends classical Viskovatov method to multiple series.

Hermite-Padé polynomials can be computed iteratively with known previous polynomials.

Parallelization is possible with independent evaluations at each step.

Abstract

We propose an algorithm for producing Hermite-Pad\'e polynomials of type I for an arbitrary tuple of $m+1$ formal power series $[f_0,\dots,f_m]$, $m\geq1$, about $z=0$ ($f_j\in{\mathbb C}[[z]]$) 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 construction of Pad\'e polynomials (for $m=1$ our algorithm coincides with the Viskovatov algorithm). The algorithm proposed here is based on a recurrence relation and has the feature that all the Hermite-Pad\'e polynomials corresponding to the multiindices $(k,k,k,\dots,k,k)$, $(k+1,k,k,\dots,k,k)$, $(k+1,k+1,k,\dots,k,k),\dots$, $(k+1,k+1,k+1,\dots,k+1,k)$ are already known by the time the algorithm produces the Hermite-Pad\'e polynomials corresponding to the multiindex $(k+1,k+1,k+1,\dots,k+1,k+1)$. We show how the Hermite-Pad\'e polynomials corresponding to different multiindices can be found via this algorithm by changing appropriately the initial conditions. The algorithm can be parallelized in $m+1$ independent evaluations at each $n$th step.