A Viskovatov algorithm for Hermite-Padé polynomials

Abstract

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.