Abstract
In this paper, we consider iterative formulae with high order of convergence to solve a polynomial equation,f(z)=0. First, we derive the numerator of the Padé approximant forf(z)/f′(z) by combining Viscovatov's and Euclidean algorithms, and then calculate the zeros of the numerator so as to apply one of the zeros for the next approximation. Regardless of whether the root is simple or multiple, the convergence order of this iterative formula is always attained for arbitrary positive integerm with the Taylor polynomial of degreem for a given polynomialf(z). Since it is easy to systematically obtain formulae of different order, we can choose formulae of suitable order according to the required accuracy.
Zusammenfassung
In dieser Arbeit betrachten wir Iterationsverfahren höherer Ordnung zur Lösung einer Polynomgleichungf(z)=0. Durch Anwendung der Verfahren von Viscovatov und Euclid erhalten wir eine Approximation für den Zähler der Padé-Approximierenden vonf(z)/f′(z), und verwenden eine der Wurzeln des Zählerpolynoms für die nächste Approximation. Dieses Verfahren hat die Ordnungm sowohl für einfache als auch mehrfache Wurzeln bei Verwendung des Taylor Polynomsm-ter Ordnung. Da es leicht ist, Verfahren verschiedener Ordnung zu erhalten, können wir gemäß der geforderten Genauigkeit eine passende Ordnung des Verfahrens wählen.
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References
Tornheim, L.: Convergence of multipoint iterative methods. J. Assoc. Comput. Mach.11, 210–220 (1964).
Merz, G.: Padésche Näherungsbrüche und Iterationsverfahren höherer Ordnung. Computing3, 165–183 (1968).
Chisholm, J.: Approximation by sequences of Padé approximants in region of meromorphy. J. Math. Phys.7, 1, 39–44 (1966).
Nourein M.: Root determination by use of Padé approximants. BIT16, 291–297 (1976).
Claessens, G., Loizou, G., Wuytack, L.: Comments on a root finding method using Padé approximation. BIT17, 360–361 (1977).
Mills, W. H.: Continued fractions and linear recurrences. Math. Comp.29, 129, 173–180 (1975).
Murphy, J. A.: A class of algorithms for obtaining rational approximants to functions which are defined by power series. J. Appl. Math. Phys.28, 1121–1131 (1977).
Magnus, A.: Certain continued fractions associated with the Padé table. Math. Zeitschr.78, 361–374 (1962).
Baker, G., Graves-Morris, P.: Padé approximants I. London: Addison-Wesley 1981.
Cuyt, A., Wuytack, L.: Nonlinear methods in numerical analysis. Amsterdam: North-Holland 1987.
Bultheel, A.: Recursive algorithms for the Padé table: two approaches. In [21],, 211–230.
Householder, A.: The Padé table, the Frobenius identities, and the ad algorithm. Linear Algebra and Its Appl.4, 161–174 (1971).
Baker, G.: Recursive calculation of Padé approximants. In [21],, 83–92.
Pindor, M.: A simplified algorithm for calculating the Padé table derived from Baker and Longman schemes. J. Comp. Appl. Math.2, 4, 255–257 (1976).
Gragg, W.: The Padé table and its relation to certain algorithms of numerical analysis. SIAM Review14, 1–62 (1972).
Claessens, G.: A new look at the Padé table and the different methods for computing its elements. J. Comp. Appl. Math.1, 141–152 (1975).
Brezinski, C.: Computation of Padé approximants and continued fractions. J. Comp. Appl. Math.2, 113–123 (1976).
Graves-Morris, P.: The numerical calculation of Padé approximants, In [22],, 231–245.
Baker, G.: Essentials of Padé approximants. New York: Academic Press 1975.
McEliece, R., Shearer, J.: A property of Euclid's algorithm and an application to Padé approximation. SIAM J. Appl. Math.34, 4, 611–615 (1978).
Graves-Morris, P.: Padé approximants and their applications, Proc. Univ. Kent. Berlin: Springer 1972.
Wuytack, L.: Padé approximation and its applications, Proc. Antwerp. Berlin: Springer 1979.
Pomentale, T.: A class of iterative method for holomorphic functions. Numer Math.18, 193–203 (1971).
Sakurai, T., Torii, T., Sugiura, H.: An electrostatic interpretation iterative method for finding roots of polynomial. Trans. Inf. Process. Soc.29, 5, 447–455 (1988) (Japanese).
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Sakurai, T., Torii, T. & Sugiura, H. An iterative method for algebraic equation by Padé approximation. Computing 46, 131–141 (1991). https://doi.org/10.1007/BF02239167
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DOI: https://doi.org/10.1007/BF02239167