Abstract
Using the iterative method of Newton's type in circular arithmetic, introduced in [14], a new iterative method for finding a multiple complex zero of a polynomial is derived. This method can be regarded as a version of classical Schröder's method. Initial conditions which guarantee a safe convergence of the proposed method are stated. The increase of the computational efficiency is achieved by a combination of the complex approximation methods of Schröder's type with some interval methods. The presented algorithms are analysed in view of their efficiency and illustrated numerically in the example of a polynomial equation.
Zusammenfassung
Durch Verwendung der in [14] eingeführten iterativen Newton-Typ-Methode in zirkulärer Arithmetik wird eine neue iterative Methode zum Auffinden mehrfacher komplexer Nullstellen eines Polynoms hergeleitet. Diese Methode kann als Version der klassischen Schröder-Methode angesehen werden. Anfangsbedingungen, die eine sichere Konvergenz der vorgeschlagenen Methode garantieren, werden eingegeben. Die Steigerung der Berechnungseffizienz wird durch Kombination der komplexen Approximationsmethode vom Schröder-Typ mit einigen Intervallmethoden erreicht. Der erreichte Algorithmus wird in Hinsicht auf seine Effizienz analysiert und numerisch am Beispiel einer Polynomgleichung illustriert.
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This work was done during a visiting professorship at the University of Oldenburg funded by the Deutsche Forschungsgemeinschaft (DFG).
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Petković, M.S. Schröder-like algorithms for multiple complex zeros of a polynomial. Computing 45, 39–50 (1990). https://doi.org/10.1007/BF02250583
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DOI: https://doi.org/10.1007/BF02250583