Abstract
A Gauss-Seidel procedure is applied to increase the convergence of a basic fourth order method for finding polynomial complex zeros. Further acceleration of convergence is performed by using Newton's and Halley's corrections. It is proved that the lower bounds of theR-order of convergence for the proposed serial (single-step) methods lie between 4 and 7. Computational efficiency and numerical examples are also given.
Zusammenfassung
Ein Gauss-Seidel Verfahren wird verwendet zur Beschleunigung der Konvergenz eines Verfahrens vierter Ordnung zur Bestimmung komplexer Polynomwurzeln. Eine weitere Beschleunigung der Konvergenz wird erreicht durch Anwendung von Newton und Halley-Korrekturformeln. Es wird bewiesen, daß die unteren Schranken für dieR-Ordnung der vorgeschlagenen seriellen (Einzelschritt-) Verfahren zwischen 4 und 7 liegen. Die Effizienzzahl und ein numerisches Beispiel wird angegeben.
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Petković, M.S., Stefanović, L.V. & Marjanović, Z.M. On theR-order of some accelerated methods for the simultaneous finding of polynomial zeros. Computing 49, 349–361 (1993). https://doi.org/10.1007/BF02248695
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DOI: https://doi.org/10.1007/BF02248695