Abstract
Claessens' cross rule [8] enables simple computation of the values of the rational interpolation table if the table is normal, i.e. if the denominators in the cross rule are non-zero. In the exceptional case of a vanishing denominator a singular block is detected having certain structural properties so that some values are known without further computations. Nevertheless there remain entries which cannot be determined using only the cross rule.
In this note we introduce a simple recursive algorithm for computation of the values of neighbours of the singular block. This allows to compute entries in the rational interpolation table along antidiagonals even in the presence of singular blocks. Moreover, in the case of non-square singular blocks, we discuss a facility to monitor the stability.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Arndt, Ein verallgemeinerter Kettenbruch zur rationalen Hermite-Interpolation, Numer. Math. 36 (1980) 99–107.
G.A. Baker and P. Graves-Morris,Padé Approximants, Part II (Addison-Wesley, 1981).
B. Beckermann, The structure of the singular solution table of the M-Padé approximation problem, J. Comput. Appl. Math. 32 (1990) 3–15.
B. Beckermann, Zur Interpolation mit polynomialen Linearkombinationen beliebiger Funktionen, Thesis, Univ. Hannover (1990).
G. Claessens, On the Newton-Padé approximation problem, J. Approx. Th. 22 (1978) 150–160.
G. Claessens, On the structure of the Newton-Padé table, J. Approx. Th. 22 (1978) 304–319.
G. Claessens, A new algorithm for osculatory rational interpolation, Numer. Math. 27 (1976) 77–83.
G. Claessens, A useful identity for the rational Hermite interpolation table, Numer. Math. 29 (1978) 227–231.
F. Cordellier, Démonstration algébrique de l'extension de l'identité de Wynn aux table Padé non normal, in:Padé Approximation and its Applications, Antwerp 1979, Lecture Notes in Mathematics 765 (Springer, Berlin, 1979) pp. 257–277.
F. Cordellier, Utilisation de l'invariance homographique dans les algorithmes de losange, in:Padé Approximation and its Applications Bad Honnef 1983, Lecture Notes in Mathematics 1071 (Springer, Berlin, 1983) pp. 62–94.
F. Cordellier, Interpolation rationelle et autres questions: aspects algorithmes et numériques, Thèse d'Etat, Lille (1989).
P.R. Graves-Morris, Efficient reliable rational approximation, in:Padé Approximation and its Applications, Amsterdam 1980, Lecture Notes in Mathematics, 888 (Springer, Berlin, 1980), pp. 28–63.
P.R. Graves-Morris, Talk at the JCAM Conf., Leuven (1984), unpublished.
M.H. Gutknecht, Continued fraction associated with the Newton-Padé table, Numer. Math. 56 (1989) 547–589.
H.E. Salzer, Note on osculatory rational interpolation, Math. Comp. 16 (1962) 486–491.
M. Van Barel and A. Bultheel, A new approach to the rational interpolation problem, J. Comput. Appl. Math. 32 (1990) 281–289.
D.D. Warner, Hermite interpolation with rational functions, Ph.D. Thesis, Univ. of California, San Diego (1974).
H. Werner, A reliable method for rational interpolation, in:Padé Approximation and its Applications, Antwerp 1979, Lecture Notes in Mathematics 765 (Springer, Berlin, 1979) pp. 257–277.
Author information
Authors and Affiliations
Additional information
Dedicated to Professor G. Mühlbach on the occasion of his 50th birthday
Rights and permissions
About this article
Cite this article
Beckermann, B., Carstensen, C. A reliable modification of the cross rule for rational Hermite interpolation. Numer Algor 3, 29–44 (1992). https://doi.org/10.1007/BF02141913
Issue Date:
DOI: https://doi.org/10.1007/BF02141913