On the theory and computation of nonperfect Padé-Hermite approximants
A Padé-Hermite approximant of type \(n = (n_ 0,n_ 1, \dots, n_ k)\) is a set of \(k + 1\) polynomials \(P_ i(z)\) having degrees bounded by the \(n_ i\) and satisfying \(A_ 0(z) P_ 0(z) + \dots + A_ k (z) P_ k (z) = O(z^{n_ 0 + n_ 1 + \dots + n_ k + k})\). The \(A_ i(z)\) are \(k + 1\) given power series. The special case \(k = 1\) and \(A_ 1(z) = -1\) gives the classical Padé approximation of a single power series. Algorithms for computing Padé-Hermite approximants require the vector of the power series \(A_ i(z)\) to be perfect. This restriction is a strong one. In the present paper, a new and fast algorithm is given which can be applied to any vector of power series. Two new types of rational approximants, the weak Padé-Hermite form and the weak Padé-Hermite fraction, are introduced and discussed.
scientific article; zbMATH DE number 89593
scientific article