Abstract
The convergence of columns in the univariateqd-algorithm to reciprocals of polar singularities of meromorphic functions has often proved to be very useful. A multivariateqd-algorithm was discovered in 1982 for the construction of the so-called homogeneous Padé approximants.
In the first section we repeat the univariate convergence results. In the second section we summarize the “homogeneous” multivariateqd-algorithm. In the third section a multivariate convergence result is proved by combining results from the previous sections. This convergence result is compared with another theorem for the general order multivariateqdg-algorithm. The main difference lies in the fact that the homogeneous form detects the polar singularities “pointwise” while the general form detects them “curvewise”.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Chaffy-Camus,Interpolation polnomiale et ratioanelle d'une fonction de plusieurs variables complexes, Grenoble, Thèse 3ième cycle, 1984.
A. Cuyt,The qd-algorithm and multivariate Padé approximants, Numer. Math., 42 (1983), pp. 259–269.
A. Cuyt,Padé Approximants for Operators: Theory and Applications (LNM 1065), Springer, Berlin, 1984.
A. Cuyt,Where do the columns of the multivariate QD-algorithms go? A proof constructed with the aid of Mathematica, Preprint UIA, 93–20 (1983).
P. Henrici,Applied and Computational Complex Analysis I, J. Wiley, New York, 1974.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cuyt, A. On the convergence of the multivariate “homogeneous”qd-algorithm. BIT 34, 535–545 (1994). https://doi.org/10.1007/BF01934266
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01934266