Viskovatov algorithm for Hermite-Padé polynomials

Abstract

We propose an algorithm for producing Hermite-Padé polynomials of type I for an arbitrary tuple of m + 1 formal power series [f0, . . . , fm], m) 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 Padé 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-Padé polynomials corresponding to the multiindices (k,

Tschebyshev-Padé approximations for multivalued functions

“…it is possible to use the algorithms for computing the type I Hermite-Padé polynomials for the tuple [1, f 1 , f 2 ] and multiindex n = (n − 1, n, n); see [22], [23] and the bibliography therein. Note that both Tschebyshev-Padé approximations and Hermite-Padé polynomials are constructive in the sense of P. Henrici paper [18, Sec.…”

“…All the numerical computations were performed using the Program HePa.com [23]. This Program is based on a generalization of the classical Viskovatov algorithm [22]. Here the zeros of Padé polynomials P n,0 (z), P n,1 (z) of order n = 100 for the function f given by (42) are plotted (blue points for P n,0 and red points for P n,1 ).…”