A Viskovatov algorithm for Hermite-Padé polynomials

Abstract

We propose and justify an algorithm for producing Hermite- Padé polynomials of type I for an arbitrary tuple of formal power series , , about the point ( ) 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 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 , , , , are already known at the point when the algorithm produces the Hermite-Padé polynomials corresponding to the multi- index . 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 , the algorithm can be parallelized in independent evaluations. Bibliography: 30 titles.

Scalar Equilibrium Problem and the Limit Distribution of Zeros of Hermite-Padé Polynomials of Type II

“…Thus again based on a generalization of the Viskovatov algorithm [13], [14] and different multiindexes it is possible to compute HP-polynomials through type I HP-polynomials.…”

On Some Algebraic Properties of Hermite--Padé Polynomials

“…Under some assumption on the "general position" of the tuple (cf. [9], [3], [11]) this construction provides two (m + 1) × (m + 1) polynomial matrices M 1 (z) and M 2 (z), M 1 (z), M 2 (z) ∈ GL(m + 1, C[z]), with the following property: M 1 (z)M 2 (z) ≡ I m+1 , where I m+1 is the identity (m + 1) × (m + 1)-matrix.…”