Recursive Algorithms for Nonnormal Padé Tables

Abstract

The Berlekamp-Massey algorithm [L. S. De Jong, Numerical aspects of realization algorithms in linearsystems theory, Ph.D. thesis, T. H. Eindhoven, the Netherlands, 1975] and [J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Information Theory, IT-15 (1969), pp. 122–127] for minimal realization problems is a special case of the Padé approximation problem. As a matter of fact, it computes among other polynomials the denominators of the elements of the Padé table that are on the descending diagonal $\{ [0/1],[1/2], \cdots ,[k/k + 1], \cdots \} $ as far as they exist and this algorithm works for nonnormal Padé tables too. This algorithm does not seem to be well-known in Padé approximation literature. It is not very difficult to generalize this algorithm so as to compute the other Padé approximants of a nonnormal table. Some variants will lead to a generalization of the algorithm of Brezinski [Computation of Padé approximants continued fractions, J. Comput. Appl. Math., 2 (1976), pp. 113–123], which computes the descending diagonals of a normal Padé table, and of the algorithm of Watson [D. Bussonnais, “Tous” les algorithmes de calcul par recurrence des approximants de Padé d’une serie, Construction de fractions continues correspondantes, Séminaire d’Analyse Numérique, No. 293, Grenoble,1978], [G. Claessens, new look at the Padé table and the different methods for computing its elements, J. Comput. Appl. Math., 1 (1975), pp. 141–152], [P. J. S. Watson, Algorithms for differentiation and integration, Padé Approximants and their Applications, P. R. Groves-Morris, ed., Academic Press London, 1973, pp. 93–98] that computes the descending staircases of normal tables. This work is the dual of Cordellier’s [algorithmes de calcul recursif des elements d’une table de Padé non normale, Conference on Padé approximation, Lille, France, 1978], and of McEliece and Shearer [A property of Euclid’s algorithm and an application to Padé approximation, SIAM J. Appl. Math., 34 (1978), pp. 611–616] deriving similar results for ascending diagonals and staircases. A more continued fraction-like approach would generalize the Thatcher algorithm and may be found in [J. A. Murphy and H. R. O’Donohoe, A class of algorithms for obtaining rational approximants to functions which are defined by power series, Z. Angew. Math. Phys., 28 (1977), pp. 1121–1131] and [A. Bultheel, Division algorithms for continued fractions and the Padé table, Applied Mathematics and Programming Division, Rep. TW41, Katholicke Universitiet Leuven, Heverlee, Belgium, August1978].