Summary
The Quotient-Difference Algorithm of Rutishauser can be used for the determination of poles of a meromorphic function given by its power series. If some of the poles have same modulus, a sequence of polynomials can be determined such that the limiting polynomial has exactly these poles as zeros. The convergence has not been proved by Rutishauser, however. A proof is presented in this paper.
Zusammenfassung
Der Quotienten-Differenzen-Algorithmus nach Rutishauser ist geeignet zur Bestimmung von Polen meromorpher Funktionen, gegeben durch eine Taylorreihe. Sind mehrere Pole betragsgleich, so kann eine Polynomfolge bestimmt werden, deren Grenzpolynom diese Pole als Nullstellen hat. Die Konvergenz wurde von Rutishauser jedoch nicht bewiesen. Ein Beweis wird in der vorliegenden Arbeit präsentiert.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literatur
Henrici, P.: Some Applications of the Quotient-difference Algorithm, Proc. Symp. Appl. Math. Amer. Math. Soc.15, 159–183 (1963)
Henrici, P.: Quotient-Difference Algorithms. In: Mathematical Methods for Digital Computers, Vol. II. Ralston, A., Wilf, H.S. (eds.), New York: John Wiley and Sons 1967
Henrici, P.: Applied and Computational Complex Analysis, Vol. I, § 7. New York: John Wiley and Sons 1974
Householder, A.S.: The Numerical Treatment of a Single Nonlinear Equation. New York: McGraw-Hill Book Company, 1970
Householder, A.S.: The Padé Table, the Frobenius Identities, and the QD Algorithm. In: Linear Algebra and Appl.4, 161–174 (1971)
Rutishauser, H.: Der Quotienten-Differenzen-Algorithmus. ZAMP5, 233–251 (1954)
Rutishauser, H.: Der Quotienten-Differenzen-Algorithmus, Mitt. Nr. 7. Institut f. angew. Math. der ETH Zürich, Birkhäuser, Basel, 1957
Stewart, G.W.: On a Companion Operator for Analytic Functions. Numer. Math.18, 26–43 (1971)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Seewald, W. Quotienten-Differenzen-Algorithmus: Beweis der Regeln von Rutishauser. Numer. Math. 40, 93–98 (1982). https://doi.org/10.1007/BF01459078
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01459078