On the theory and computation of nonperfect padé-hermite approximants

AbstractFor a vector of k + 1 power series we introduce two new types of rational approximations, the weak Padé-Hermite form and the weak Padé-Hermite fraction. A recurrence relation is then presented which computes Padé-Hermite forms along with their weak counterparts along a sequence of perfect points in the Padé-Hermite table. The recurrence relation results in a fast algorithm for calculating a Padé-Hermite approximant of any given type. When the vector of power series is perfect, the algorithm is shown to calculate a Padé-hermite form of type (n0,…,nk) in O(kN)2 operations, where N = n0 + ⋯ + nk. This complexity is the same as that of other fast algorithms. The new algorithm also succeeds in the nonperfect case, usually with only a moderate increase in cost