On the theory and computation of nonperfect Pade´-Hermite approximants

Abstract For a vector of k + 1 power series we introduce two new types of rational approximations, the weak Pade-Hermite form and the weak Pade-Hermite fraction. A recurrence relation is then presented which computes Pade-Hermite forms along with their weak counterparts along a sequence of perfect points in the Pade-Hermite table. The recurrence relation results in a fast algorithm for calculating a Pade-Hermite approximant of any given type. When the vector of power series is perfect, the algorithm is shown to calculate a Pade-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.

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