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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Alefeld, G., Herzberger, J.: Introduction to Interval Computation. New York: Academic Press 1983.
Gargantini, I.: Parallel Laguerre iterations: Complex case. Numer. Math.26, 317–323 (1976).
Gargantini, I.: Further applications of circular arithmetic: Schroeder-like algorithms with error bounds for finding zeros of polynomials. SIAM J. Numer. Anal.15, 497–510 (1978).
Gargantini, I., Henrici, P.: Circular arithmetic and the determination of polynomial zeros. Numer. Math.18, 305–320 (1972).
Hansen, E., Patrick, M.: A family of root finding methods. Numer. Math.27, 257–269 (1977).
Henrici, P.: Applied and Computational Complex Analysis Vol. I. New York: John Wiley and Sons Inc. 1974.
Kronsjö, L. I.: Algorithms: Their Complexity and Efficiency. Chichester: John Wiley and Sons Inc. 1979.
Lagouanelle, J. L.: Sur une métode de calcul de l'ordre de multiplicité des zéros d'un polynôme. C. R. Acad. Sci. Paris Sér. A.262, 626–627 (1966).
Marden, M.: Geometry of Polynomials. Providence: Amer. Math. Soc. 1966.
Moore, R. E.: Interval Analysis. New Jersey: Prentice-Hall 1966.
Nesdoré, P. F.: The determination of an algorithm which uses the mixed strategy thechnique for the solution of single nonlinear equations, in: Numerical Methods for Nonlinear Algebraic Equations (ed. P. Rabinowitz). London: Gordon and Breach Science Publishers 1970, p. 27–45.
Petković, M. S.: On a generalisation of the root iterations for polynomial complex zeros in circular interval arithmetic. Computing27, 37–55 (1981).
Petković, M. S.: On an iterative method for simultaneous inclusion of polynomial complex zeros. J. Comput. Appl. Math.8, 51–56 (1982).
Petković, M. S.: On some interval iterations for finding a zero of a polynomial with error bounds. Comput. Math. Appl.14, 479–495 (1987).
Petković, M. S.: On the Halley-like algorithms for the simultaneous approximation of polynomial complex zeros. SIAM J. Numer. Anal.26, 740–753 (1989).
Petković, M. S.: Iterative Methods for Simultaneous Inclusion of Polynomial Zeros. Berlin: Springer-Verlag 1989.
Petković, M. S.: On the efficiency of some combined methods for polynomial zeros. J. Comput. Appl. Math.30, 99–115 (1990).
Petković, M. S., Stefanović, L. V.: On the simultaneous method of the second order for finding polynomial complex zeros in circular arithmetic. Freiburger Intervall-Berichte3, 63–95 (1985).
Schröder, E.: Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann.2, 317–365 (1870).
Traub, J. F.: Iterative Methods for the Solution of Equations. New Jersey: Prentice-Hall 1964.
Wang, X., Zheng, S.: A family of parallel and interval iterations for finding all roots of a polynomial simultaneously with rapid convergence (II) (in Chinese). J. Comput. Math.4, 433–444 (1985).
Wang, X., Zheng, S.: The quasi-Newton method in parallel circular iteration. J. Comput. Math.4, 305–309 (1984).
Wang, X., Zheng, S., Shen, G.: Bell's disk polynomial and parallel disk iteration. Numer. Math., J. Chinese Univ.4, 328–345 (1987).
Wilkinson, J. H.: Rounding Errors in Algebraic Processes. New Jersey: Prentice-Hall 1963.
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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