Abstract
In one of his papers [5] Gautschi presents an algorithm for determining the minimal solution of a second-order homogeneous difference equation. The method is based on the connection between the existence of a minimal solution of such a difference equation and the convergence of a certain continued fraction. In the present paper, these results are generalized. For this purpose we use the concept of generalized continued fraction. The resulting algorithm is suitable for solving (n+1)-th order recursions (n≥1) for which there existn independent solutions that are dominated by each solution that does not belong to the space spanned by thesen solutions.
Zusammenfassung
In einer seiner Arbeiten [5] gibt Gautschi einen Algorithmus an, um die minimale Lösung einer homogenen Differenzengleichung 2. Ordnung zu bestimmen. Die Methode basiert auf dem Zusammenhang zwischen der Existenz einer minimalen Lösung einer solchen Differenzengleichung und der Konvergenz eines gewissen Kettenbruchs. Diese Ergebnisse werden hier verallgemeinert. Dazu benutzen wir das Konzept eines verallgemeinerten Kettenbruchs. Der Algorithmus ist geeignet, um Rekursionen der Ordnung (n+1) zu lösen (n≥1), für dien linear unabhängige Lösungen existieren, die von jeder Lösung dominiert werden, die nicht zu dem von ihnen aufgespannten Unterraum gehört.
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Van der Cruyssen, P. Linear difference equations and generalized continued fractions. Computing 22, 269–278 (1979). https://doi.org/10.1007/BF02243567
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DOI: https://doi.org/10.1007/BF02243567